FAILURE MAP
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FA-69621 / Orbital propagation / Open access

Stumpff functions for universal variables: Hyperbolic S uses the elliptic numerator order · case 01

Time of flight on hyperbolic arcs is negative.

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

ROOT CAUSE

The numerator is s - sinh s.

THE FAILURE

The numerator is s - sinh s.

Unsuccessful approach: Fixing the sign but dividing by s^2 mis-scales S.

Case contract

Input z. For z>1e-6: C=(1-cos s)/z, S=(s-sin s)/s^3 with s=sqrt z; for z<-1e-6: C=(cosh s-1)/(-z), S=(sinh s-s)/s^3 with s=sqrt(-z); otherwise series C=1/2-z/24+z^2/720, S=1/6-z/120+z^2/5040. Return [C,S] rounded to 10.

Why this case matters

Orbit determination and mission planning chain many small conversions; one wrong branch or unit silently moves a spacecraft by kilometres.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    z=x
    if z>1e-6:
        s=math.sqrt(z)
        c=(1-math.cos(s))/z; S=(s-math.sin(s))/s**3
    elif z<-1e-6:
        s=math.sqrt(-z)
        c=(math.cosh(s)-1)/(-z); S=(s-math.sinh(s))/s**3
    else:
        c=1/2-z/24+z*z/720; S=1/6-z/120+z*z/5040
    return [round(c,10),round(S,10)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('stumpff functions for universal variables -0.5', -0.5, [0.521183673, 0.1708832825]), ('stumpff functions for universal variables 4.0', 4.0, [0.3540367091, 0.1363378216]), ('stumpff functions for universal variables 0.5', 0.5, [0.4795108058, 0.1625492603]), ('stumpff functions for universal variables 30.0', 30.0, [0.0102526963, 0.0377242152]), ('stumpff functions for universal variables 1e-07', 1e-07, [0.4999999958, 0.1666666658]), ('stumpff functions for universal variables 0.0', 0.0, [0.5, 0.1666666667]), ('stumpff functions for universal variables -1e-07', -1e-07, [0.5000000042, 0.1666666675])], [('stumpff functions for universal variables -4.0', -4.0, [0.6905489228, 0.203357551]), ('stumpff functions for universal variables 1e-07', 1e-07, [0.4999999958, 0.1666666658]), ('stumpff functions for universal variables 0.0', 0.0, [0.5, 0.1666666667]), ('stumpff functions for universal variables -1e-07', -1e-07, [0.5000000042, 0.1666666675]), ('stumpff functions for universal variables -0.5', -0.5, [0.521183673, 0.1708832825]), ('stumpff functions for universal variables -25.0', -25.0, [2.928397941, 0.5536256846]), ('stumpff functions for universal variables 9.8696', 9.8696, [0.2026424576, 0.1013212062])], [('stumpff functions for universal variables -25.0', -25.0, [2.928397941, 0.5536256846]), ('stumpff functions for universal variables -0.5', -0.5, [0.521183673, 0.1708832825]), ('stumpff functions for universal variables -4.0', -4.0, [0.6905489228, 0.203357551]), ('stumpff functions for universal variables 9.8696', 9.8696, [0.2026424576, 0.1013212062]), ('stumpff functions for universal variables 0.05', 0.05, [0.4979201358, 0.1662504957]), ('stumpff functions for universal variables -0.05', -0.05, [0.5020868087, 0.1670838297]), ('stumpff functions for universal variables 2e-06', 2e-06, [0.4999999167, 0.16666665])], [('stumpff functions for universal variables -0.05', -0.05, [0.5020868087, 0.1670838297]), ('stumpff functions for universal variables 9.8696', 9.8696, [0.2026424576, 0.1013212062]), ('stumpff functions for universal variables 0.05', 0.05, [0.4979201358, 0.1662504957]), ('stumpff functions for universal variables 2e-06', 2e-06, [0.4999999167, 0.16666665]), ('stumpff functions for universal variables -3e-06', -3e-06, [0.500000125, 0.1666666917]), ('stumpff functions for universal variables 4.0', 4.0, [0.3540367091, 0.1363378216]), ('stumpff functions for universal variables 0.5', 0.5, [0.4795108058, 0.1625492603])], [('stumpff functions for universal variables -3e-06', -3e-06, [0.500000125, 0.1666666917]), ('stumpff functions for universal variables 2e-06', 2e-06, [0.4999999167, 0.16666665]), ('stumpff functions for universal variables 4.0', 4.0, [0.3540367091, 0.1363378216]), ('stumpff functions for universal variables 0.5', 0.5, [0.4795108058, 0.1625492603]), ('stumpff functions for universal variables 30.0', 30.0, [0.0102526963, 0.0377242152]), ('stumpff functions for universal variables 1e-07', 1e-07, [0.4999999958, 0.1666666658]), ('stumpff functions for universal variables 0.0', 0.0, [0.5, 0.1666666667])]]
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 fixtureActualExpectedOutcome
stumpff functions for universal variables -0.5[0.521183673, -0.1708832825][0.521183673, 0.1708832825]Failed
stumpff functions for universal variables 4.0[0.3540367091, 0.1363378216][0.3540367091, 0.1363378216]Passed
stumpff functions for universal variables 0.5[0.4795108058, 0.1625492603][0.4795108058, 0.1625492603]Passed
stumpff functions for universal variables 30.0[0.0102526963, 0.0377242152][0.0102526963, 0.0377242152]Passed
stumpff functions for universal variables 1e-07[0.4999999958, 0.1666666658][0.4999999958, 0.1666666658]Passed
stumpff functions for universal variables 0.0[0.5, 0.1666666667][0.5, 0.1666666667]Passed
stumpff functions for universal variables -1e-07[0.5000000042, 0.1666666675][0.5000000042, 0.1666666675]Passed

SHA-256 / da1a5bc33f25ca07f00b8f32174af82435ab1f672785204dc6eddf11cb880a88

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    z=x
    if z>1e-6:
        s=math.sqrt(z)
        c=(1-math.cos(s))/z; S=(s-math.sin(s))/s**3
    elif z<-1e-6:
        s=math.sqrt(-z)
        c=(math.cosh(s)-1)/(-z); S=(math.sinh(s)-s)/s**2
    else:
        c=1/2-z/24+z*z/720; S=1/6-z/120+z*z/5040
    return [round(c,10),round(S,10)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('stumpff functions for universal variables -0.5', -0.5, [0.521183673, 0.1708832825]), ('stumpff functions for universal variables 4.0', 4.0, [0.3540367091, 0.1363378216]), ('stumpff functions for universal variables 0.5', 0.5, [0.4795108058, 0.1625492603]), ('stumpff functions for universal variables 30.0', 30.0, [0.0102526963, 0.0377242152]), ('stumpff functions for universal variables 1e-07', 1e-07, [0.4999999958, 0.1666666658]), ('stumpff functions for universal variables 0.0', 0.0, [0.5, 0.1666666667]), ('stumpff functions for universal variables -1e-07', -1e-07, [0.5000000042, 0.1666666675])], [('stumpff functions for universal variables -4.0', -4.0, [0.6905489228, 0.203357551]), ('stumpff functions for universal variables 1e-07', 1e-07, [0.4999999958, 0.1666666658]), ('stumpff functions for universal variables 0.0', 0.0, [0.5, 0.1666666667]), ('stumpff functions for universal variables -1e-07', -1e-07, [0.5000000042, 0.1666666675]), ('stumpff functions for universal variables -0.5', -0.5, [0.521183673, 0.1708832825]), ('stumpff functions for universal variables -25.0', -25.0, [2.928397941, 0.5536256846]), ('stumpff functions for universal variables 9.8696', 9.8696, [0.2026424576, 0.1013212062])], [('stumpff functions for universal variables -25.0', -25.0, [2.928397941, 0.5536256846]), ('stumpff functions for universal variables -0.5', -0.5, [0.521183673, 0.1708832825]), ('stumpff functions for universal variables -4.0', -4.0, [0.6905489228, 0.203357551]), ('stumpff functions for universal variables 9.8696', 9.8696, [0.2026424576, 0.1013212062]), ('stumpff functions for universal variables 0.05', 0.05, [0.4979201358, 0.1662504957]), ('stumpff functions for universal variables -0.05', -0.05, [0.5020868087, 0.1670838297]), ('stumpff functions for universal variables 2e-06', 2e-06, [0.4999999167, 0.16666665])], [('stumpff functions for universal variables -0.05', -0.05, [0.5020868087, 0.1670838297]), ('stumpff functions for universal variables 9.8696', 9.8696, [0.2026424576, 0.1013212062]), ('stumpff functions for universal variables 0.05', 0.05, [0.4979201358, 0.1662504957]), ('stumpff functions for universal variables 2e-06', 2e-06, [0.4999999167, 0.16666665]), ('stumpff functions for universal variables -3e-06', -3e-06, [0.500000125, 0.1666666917]), ('stumpff functions for universal variables 4.0', 4.0, [0.3540367091, 0.1363378216]), ('stumpff functions for universal variables 0.5', 0.5, [0.4795108058, 0.1625492603])], [('stumpff functions for universal variables -3e-06', -3e-06, [0.500000125, 0.1666666917]), ('stumpff functions for universal variables 2e-06', 2e-06, [0.4999999167, 0.16666665]), ('stumpff functions for universal variables 4.0', 4.0, [0.3540367091, 0.1363378216]), ('stumpff functions for universal variables 0.5', 0.5, [0.4795108058, 0.1625492603]), ('stumpff functions for universal variables 30.0', 30.0, [0.0102526963, 0.0377242152]), ('stumpff functions for universal variables 1e-07', 1e-07, [0.4999999958, 0.1666666658]), ('stumpff functions for universal variables 0.0', 0.0, [0.5, 0.1666666667])]]
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 fixtureActualExpectedOutcome
stumpff functions for universal variables -0.5[0.521183673, 0.1208327279][0.521183673, 0.1708832825]Failed
stumpff functions for universal variables 4.0[0.3540367091, 0.1363378216][0.3540367091, 0.1363378216]Passed
stumpff functions for universal variables 0.5[0.4795108058, 0.1625492603][0.4795108058, 0.1625492603]Passed
stumpff functions for universal variables 30.0[0.0102526963, 0.0377242152][0.0102526963, 0.0377242152]Passed
stumpff functions for universal variables 1e-07[0.4999999958, 0.1666666658][0.4999999958, 0.1666666658]Passed
stumpff functions for universal variables 0.0[0.5, 0.1666666667][0.5, 0.1666666667]Passed
stumpff functions for universal variables -1e-07[0.5000000042, 0.1666666675][0.5000000042, 0.1666666675]Passed

SHA-256 / 1709ebdb70dee04a3778b6635bb296c20a3b755a7d9ab895dc55a3f5871fa9cf

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic toy two-body model with stipulated constants and conventions; not flight dynamics software or a validated SGP4 implementation. 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 2026-09-29T14:48:13.198188+00:00.

Case digest / 0c000f0c9879674a1b301fe41106e69f6ba8003638d61075349fe908d44b5242