Average Error: 0.5 → 0.6
Time: 23.0s
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 r6996838 = 1.0;
        double r6996839 = 5.0;
        double r6996840 = v;
        double r6996841 = r6996840 * r6996840;
        double r6996842 = r6996839 * r6996841;
        double r6996843 = r6996838 - r6996842;
        double r6996844 = r6996841 - r6996838;
        double r6996845 = r6996843 / r6996844;
        double r6996846 = acos(r6996845);
        return r6996846;
}
↓
double f(double v) {
        double r6996847 = 1.0;
        double r6996848 = v;
        double r6996849 = r6996848 * r6996848;
        double r6996850 = 5.0;
        double r6996851 = r6996849 * r6996850;
        double r6996852 = r6996847 - r6996851;
        double r6996853 = r6996849 - r6996847;
        double r6996854 = r6996852 / r6996853;
        double r6996855 = r6996854 * r6996854;
        double r6996856 = r6996854 * r6996855;
        double r6996857 = cbrt(r6996856);
        double r6996858 = acos(r6996857);
        return r6996858;
}

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))))