Average Error: 0.6 → 0.6
Time: 16.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}\right)\right)}
double f(double v) {
        double r184849 = 1.0;
        double r184850 = 5.0;
        double r184851 = v;
        double r184852 = r184851 * r184851;
        double r184853 = r184850 * r184852;
        double r184854 = r184849 - r184853;
        double r184855 = r184852 - r184849;
        double r184856 = r184854 / r184855;
        double r184857 = acos(r184856);
        return r184857;
}
↓
double f(double v) {
        double r184858 = 1.0;
        double r184859 = 5.0;
        double r184860 = v;
        double r184861 = 2.0;
        double r184862 = pow(r184860, r184861);
        double r184863 = r184859 * r184862;
        double r184864 = r184858 - r184863;
        double r184865 = r184862 - r184858;
        double r184866 = r184864 / r184865;
        double r184867 = acos(r184866);
        double r184868 = 3.0;
        double r184869 = pow(r184867, r184868);
        double r184870 = log1p(r184869);
        double r184871 = expm1(r184870);
        double r184872 = cbrt(r184871);
        return r184872;
}

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-cbrt-cube1.5

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

  4. Simplified1.5

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.6

    \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}\right)\right)}}\]

  7. Final simplification0.6

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}\right)\right)}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))