Average Error: 0.6 → 0.6
Time: 1.2m
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)\right)\]
\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 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)\right)
double f(double v) {
        double r7010884 = 1.0;
        double r7010885 = 5.0;
        double r7010886 = v;
        double r7010887 = r7010886 * r7010886;
        double r7010888 = r7010885 * r7010887;
        double r7010889 = r7010884 - r7010888;
        double r7010890 = r7010887 - r7010884;
        double r7010891 = r7010889 / r7010890;
        double r7010892 = acos(r7010891);
        return r7010892;
}
↓
double f(double v) {
        double r7010893 = 1.0;
        double r7010894 = v;
        double r7010895 = r7010894 * r7010894;
        double r7010896 = 5.0;
        double r7010897 = r7010895 * r7010896;
        double r7010898 = r7010893 - r7010897;
        double r7010899 = r7010895 - r7010893;
        double r7010900 = r7010898 / r7010899;
        double r7010901 = acos(r7010900);
        double r7010902 = log1p(r7010901);
        double r7010903 = expm1(r7010902);
        return r7010903;
}

Error

Bits error versus v

Try it out

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

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))