Average Error: 0.5 → 0.6
Time: 22.8s
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 r6996828 = 1.0;
        double r6996829 = 5.0;
        double r6996830 = v;
        double r6996831 = r6996830 * r6996830;
        double r6996832 = r6996829 * r6996831;
        double r6996833 = r6996828 - r6996832;
        double r6996834 = r6996831 - r6996828;
        double r6996835 = r6996833 / r6996834;
        double r6996836 = acos(r6996835);
        return r6996836;
}
↓
double f(double v) {
        double r6996837 = 1.0;
        double r6996838 = v;
        double r6996839 = r6996838 * r6996838;
        double r6996840 = 5.0;
        double r6996841 = r6996839 * r6996840;
        double r6996842 = r6996837 - r6996841;
        double r6996843 = r6996839 - r6996837;
        double r6996844 = r6996842 / r6996843;
        double r6996845 = r6996844 * r6996844;
        double r6996846 = r6996844 * r6996845;
        double r6996847 = cbrt(r6996846);
        double r6996848 = acos(r6996847);
        return r6996848;
}

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 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))