Average Error: 0.5 → 0.6
Time: 6.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt[3]{v \cdot v - 1}}\right)\right)\right)\]
\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}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt[3]{v \cdot v - 1}}\right)\right)\right)
double f(double v) {
        double r278961 = 1.0;
        double r278962 = 5.0;
        double r278963 = v;
        double r278964 = r278963 * r278963;
        double r278965 = r278962 * r278964;
        double r278966 = r278961 - r278965;
        double r278967 = r278964 - r278961;
        double r278968 = r278966 / r278967;
        double r278969 = acos(r278968);
        return r278969;
}
↓
double f(double v) {
        double r278970 = 1.0;
        double r278971 = v;
        double r278972 = r278971 * r278971;
        double r278973 = 1.0;
        double r278974 = r278972 - r278973;
        double r278975 = cbrt(r278974);
        double r278976 = r278975 * r278975;
        double r278977 = r278970 / r278976;
        double r278978 = 5.0;
        double r278979 = r278978 * r278972;
        double r278980 = r278973 - r278979;
        double r278981 = r278980 / r278975;
        double r278982 = r278977 * r278981;
        double r278983 = acos(r278982);
        double r278984 = log1p(r278983);
        double r278985 = expm1(r278984);
        return r278985;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  3. Applied add-cube-cbrt0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}\right) \cdot \sqrt[3]{v \cdot v - 1}}}\right)\]

  4. Applied *-un-lft-identity0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)}}{\left(\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}\right) \cdot \sqrt[3]{v \cdot v - 1}}\right)\]

  5. Applied times-frac0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt[3]{v \cdot v - 1}}\right)}\]
  6. Using strategy rm

  7. Applied expm1-log1p-u0.6

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt[3]{v \cdot v - 1}}\right)\right)\right)}\]

  8. Final simplification0.6

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt[3]{v \cdot v - 1}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))