Average Error: 0.6 → 0.6
Time: 5.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\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)\right)}
double f(double v) {
        double r339956 = 1.0;
        double r339957 = 5.0;
        double r339958 = v;
        double r339959 = r339958 * r339958;
        double r339960 = r339957 * r339959;
        double r339961 = r339956 - r339960;
        double r339962 = r339959 - r339956;
        double r339963 = r339961 / r339962;
        double r339964 = acos(r339963);
        return r339964;
}
↓
double f(double v) {
        double r339965 = 1.0;
        double r339966 = 5.0;
        double r339967 = v;
        double r339968 = r339967 * r339967;
        double r339969 = r339966 * r339968;
        double r339970 = r339965 - r339969;
        double r339971 = r339968 - r339965;
        double r339972 = r339970 / r339971;
        double r339973 = acos(r339972);
        double r339974 = log1p(r339973);
        double r339975 = expm1(r339974);
        double r339976 = log(r339975);
        double r339977 = exp(r339976);
        return r339977;
}

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 add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]
  4. Using strategy rm

  5. Applied expm1-log1p-u0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))