FA-68891 / Tide and marine navigation tables / Open access

Cosine curve time for height: Inverse cosine uses the rising-branch complement · case 01

Heights near low water are reached at the wrong end of the half-cycle.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The inversion uses acos(2*frac-1), i.e. time measured back from the second turning point.

VERIFIED REPAIR

Invert with acos(1-2*frac).

Unsuccessful approach: Using asin(2*frac-1) shifts the result by a quarter cycle.

Case contract

Input [t1,h1,t2,h2,h] (hours, metres) for consecutive turning points with t2>t1 and h1!=h2. Height follows h1+(h2-h1)*(1-cos(pi*s/T))/2 with s elapsed and T=t2-t1. Return the time in whole minutes (rounded half up) when the height h is first reached on that half-cycle, or None when h lies outside the closed range of h1..h2.

Why this case matters

Tide tables and passage plans turn published predictions into go/no-go decisions about depth, clearance and timing.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    t1,h1,t2,h2,h=x
    lo,hi=min(h1,h2),max(h1,h2)
    if h<lo or h>hi: return None
    frac=(h-h1)/(h2-h1)
    T=t2-t1
    s=T*math.acos(2*frac-1)/math.pi
    return math.floor((t1+s)*60+0.5)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]', [0, 0.6, 6.2, 4.4, 0.6], 0), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]', [0, 0.6, 6.2, 4.4, 2.5], 186), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]', [0, 0.6, 6.2, 4.4, 4.5], None), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]', [0, 0.6, 6.2, 4.4, 4.5], None), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 4.0]', [3.5, 5.1, 9.75, 1.2, 4.0], 344), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 3.15]', [3.5, 5.1, 9.75, 1.2, 3.15], 398)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]', [0, 0.6, 6.2, 4.4, 2.5], 186), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 4.0]', [3.5, 5.1, 9.75, 1.2, 4.0], 344), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 3.15]', [3.5, 5.1, 9.75, 1.2, 3.15], 398), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.3]', [3.5, 5.1, 9.75, 1.2, 1.3], 547), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.2]', [3.5, 5.1, 9.75, 1.2, 1.2], 585)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.3]', [3.5, 5.1, 9.75, 1.2, 1.3], 547), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.2]', [3.5, 5.1, 9.75, 1.2, 1.2], 585), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 0.9]', [3.5, 5.1, 9.75, 1.2, 0.9], None), ('cosine curve time for height [10, 2.0, 22.4, 2.6, 2.3]', [10, 2.0, 22.4, 2.6, 2.3], 972), ('cosine curve time for height [0, 0, 2, 1, 0.25]', [0, 0, 2, 1, 0.25], 40)], [('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [10, 2.0, 22.4, 2.6, 2.3]', [10, 2.0, 22.4, 2.6, 2.3], 972), ('cosine curve time for height [0, 0, 2, 1, 0.25]', [0, 0, 2, 1, 0.25], 40), ('cosine curve time for height [1.25, 1.0, 7.25, 3.0, 2.0]', [1.25, 1.0, 7.25, 3.0, 2.0], 255), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]', [0, 0.6, 6.2, 4.4, 0.6], 0), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78)]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]	`372`	`0`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]	`294`	`78`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]	`186`	`186`	Passed
cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]	`88`	`284`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]	`0`	`372`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]	`None`	`None`	Passed
cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]	`585`	`210`	Failed

SHA-256 / 5539be11db2179e4f5e8a5962ded2c941abd470886851994e0d4e70e8efab9d4

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    t1,h1,t2,h2,h=x
    lo,hi=min(h1,h2),max(h1,h2)
    if h<lo or h>hi: return None
    frac=(h-h1)/(h2-h1)
    T=t2-t1
    s=T*math.asin(2*frac-1)/math.pi
    return math.floor((t1+s)*60+0.5)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]', [0, 0.6, 6.2, 4.4, 0.6], 0), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]', [0, 0.6, 6.2, 4.4, 2.5], 186), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]', [0, 0.6, 6.2, 4.4, 4.5], None), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]', [0, 0.6, 6.2, 4.4, 4.5], None), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 4.0]', [3.5, 5.1, 9.75, 1.2, 4.0], 344), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 3.15]', [3.5, 5.1, 9.75, 1.2, 3.15], 398)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]', [0, 0.6, 6.2, 4.4, 2.5], 186), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 4.0]', [3.5, 5.1, 9.75, 1.2, 4.0], 344), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 3.15]', [3.5, 5.1, 9.75, 1.2, 3.15], 398), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.3]', [3.5, 5.1, 9.75, 1.2, 1.3], 547), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.2]', [3.5, 5.1, 9.75, 1.2, 1.2], 585)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.3]', [3.5, 5.1, 9.75, 1.2, 1.3], 547), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.2]', [3.5, 5.1, 9.75, 1.2, 1.2], 585), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 0.9]', [3.5, 5.1, 9.75, 1.2, 0.9], None), ('cosine curve time for height [10, 2.0, 22.4, 2.6, 2.3]', [10, 2.0, 22.4, 2.6, 2.3], 972), ('cosine curve time for height [0, 0, 2, 1, 0.25]', [0, 0, 2, 1, 0.25], 40)], [('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [10, 2.0, 22.4, 2.6, 2.3]', [10, 2.0, 22.4, 2.6, 2.3], 972), ('cosine curve time for height [0, 0, 2, 1, 0.25]', [0, 0, 2, 1, 0.25], 40), ('cosine curve time for height [1.25, 1.0, 7.25, 3.0, 2.0]', [1.25, 1.0, 7.25, 3.0, 2.0], 255), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]', [0, 0.6, 6.2, 4.4, 0.6], 0), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78)]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]	`-186`	`0`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]	`-108`	`78`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]	`0`	`186`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]	`98`	`284`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]	`186`	`372`	Failed
cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]	`None`	`None`	Passed
cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]	`23`	`210`	Failed

SHA-256 / f479f2aa8dc7f36358105c6e936243cdf8f7385f5cc1ddffcb0a90e2f7bf2509

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    t1,h1,t2,h2,h=x
    lo,hi=min(h1,h2),max(h1,h2)
    if h<lo or h>hi: return None
    frac=(h-h1)/(h2-h1)
    T=t2-t1
    s=T*math.acos(1-2*frac)/math.pi
    return math.floor((t1+s)*60+0.5)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]', [0, 0.6, 6.2, 4.4, 0.6], 0), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]', [0, 0.6, 6.2, 4.4, 2.5], 186), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]', [0, 0.6, 6.2, 4.4, 4.5], None), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]', [0, 0.6, 6.2, 4.4, 4.5], None), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 4.0]', [3.5, 5.1, 9.75, 1.2, 4.0], 344), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 3.15]', [3.5, 5.1, 9.75, 1.2, 3.15], 398)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]', [0, 0.6, 6.2, 4.4, 2.5], 186), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 4.0]', [3.5, 5.1, 9.75, 1.2, 4.0], 344), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 3.15]', [3.5, 5.1, 9.75, 1.2, 3.15], 398), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.3]', [3.5, 5.1, 9.75, 1.2, 1.3], 547), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.2]', [3.5, 5.1, 9.75, 1.2, 1.2], 585)], [('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]', [0, 0.6, 6.2, 4.4, 3.9], 284), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.3]', [3.5, 5.1, 9.75, 1.2, 1.3], 547), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 1.2]', [3.5, 5.1, 9.75, 1.2, 1.2], 585), ('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 0.9]', [3.5, 5.1, 9.75, 1.2, 0.9], None), ('cosine curve time for height [10, 2.0, 22.4, 2.6, 2.3]', [10, 2.0, 22.4, 2.6, 2.3], 972), ('cosine curve time for height [0, 0, 2, 1, 0.25]', [0, 0, 2, 1, 0.25], 40)], [('cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]', [3.5, 5.1, 9.75, 1.2, 5.1], 210), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]', [0, 0.6, 6.2, 4.4, 4.4], 372), ('cosine curve time for height [10, 2.0, 22.4, 2.6, 2.3]', [10, 2.0, 22.4, 2.6, 2.3], 972), ('cosine curve time for height [0, 0, 2, 1, 0.25]', [0, 0, 2, 1, 0.25], 40), ('cosine curve time for height [1.25, 1.0, 7.25, 3.0, 2.0]', [1.25, 1.0, 7.25, 3.0, 2.0], 255), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]', [0, 0.6, 6.2, 4.4, 0.6], 0), ('cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]', [0, 0.6, 6.2, 4.4, 1.0], 78)]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
cosine curve time for height [0, 0.6, 6.2, 4.4, 0.6]	`0`	`0`	Passed
cosine curve time for height [0, 0.6, 6.2, 4.4, 1.0]	`78`	`78`	Passed
cosine curve time for height [0, 0.6, 6.2, 4.4, 2.5]	`186`	`186`	Passed
cosine curve time for height [0, 0.6, 6.2, 4.4, 3.9]	`284`	`284`	Passed
cosine curve time for height [0, 0.6, 6.2, 4.4, 4.4]	`372`	`372`	Passed
cosine curve time for height [0, 0.6, 6.2, 4.4, 4.5]	`None`	`None`	Passed
cosine curve time for height [3.5, 5.1, 9.75, 1.2, 5.1]	`210`	`210`	Passed

SHA-256 / 5b6fd93f09cbb873419c7ffd885dcdc508cfee29bd14d0eccef187c3974562da

Verification & scope

A deterministic toy model with stipulated rules and constants; not certified hydrographic software or a substitute for official tide tables. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:06.424294+00:00.

Case digest / f1663e59420d9816cb1936350a690c6d0cb97b80d23fb60a2d67d267c63e30da