Average Error: 0.5 → 0.6
Time: 20.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt[3]{\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}\right)
double f(double v) {
        double r8835712 = 1.0;
        double r8835713 = 5.0;
        double r8835714 = v;
        double r8835715 = r8835714 * r8835714;
        double r8835716 = r8835713 * r8835715;
        double r8835717 = r8835712 - r8835716;
        double r8835718 = r8835715 - r8835712;
        double r8835719 = r8835717 / r8835718;
        double r8835720 = acos(r8835719);
        return r8835720;
}
↓
double f(double v) {
        double r8835721 = 1.0;
        double r8835722 = v;
        double r8835723 = r8835722 * r8835722;
        double r8835724 = 5.0;
        double r8835725 = r8835723 * r8835724;
        double r8835726 = r8835721 - r8835725;
        double r8835727 = r8835723 - r8835721;
        double r8835728 = r8835726 / r8835727;
        double r8835729 = r8835728 * r8835728;
        double r8835730 = r8835728 * r8835729;
        double r8835731 = cbrt(r8835730);
        double r8835732 = acos(r8835731);
        return r8835732;
}

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-cbrt-cube0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))