Average Error: 0.6 → 0.6
Time: 23.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)
double f(double v) {
        double r187784 = 1.0;
        double r187785 = 5.0;
        double r187786 = v;
        double r187787 = r187786 * r187786;
        double r187788 = r187785 * r187787;
        double r187789 = r187784 - r187788;
        double r187790 = r187787 - r187784;
        double r187791 = r187789 / r187790;
        double r187792 = acos(r187791);
        return r187792;
}
↓
double f(double v) {
        double r187793 = 1.0;
        double r187794 = 5.0;
        double r187795 = v;
        double r187796 = 2.0;
        double r187797 = pow(r187795, r187796);
        double r187798 = r187794 * r187797;
        double r187799 = r187793 - r187798;
        double r187800 = r187797 - r187793;
        double r187801 = r187799 / r187800;
        double r187802 = acos(r187801);
        return r187802;
}

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. Taylor expanded around 0 0.6

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

  5. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))