Average Error: 0.6 → 0.6
Time: 7.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)
double f(double v) {
        double r305329 = 1.0;
        double r305330 = 5.0;
        double r305331 = v;
        double r305332 = r305331 * r305331;
        double r305333 = r305330 * r305332;
        double r305334 = r305329 - r305333;
        double r305335 = r305332 - r305329;
        double r305336 = r305334 / r305335;
        double r305337 = acos(r305336);
        return r305337;
}
↓
double f(double v) {
        double r305338 = 1.0;
        double r305339 = 5.0;
        double r305340 = v;
        double r305341 = r305340 * r305340;
        double r305342 = r305339 * r305341;
        double r305343 = r305338 - r305342;
        double r305344 = r305341 - r305338;
        double r305345 = r305343 / r305344;
        double r305346 = acos(r305345);
        double r305347 = log1p(r305346);
        double r305348 = expm1(r305347);
        return r305348;
}

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.6

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))