Average Error: 0.5 → 0.5
Time: 47.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \sqrt[3]{\left(v \cdot v\right) \cdot 5} \cdot \left(\sqrt[3]{\left(v \cdot v\right) \cdot 5} \cdot \sqrt[3]{\left(v \cdot v\right) \cdot 5}\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \sqrt[3]{\left(v \cdot v\right) \cdot 5} \cdot \left(\sqrt[3]{\left(v \cdot v\right) \cdot 5} \cdot \sqrt[3]{\left(v \cdot v\right) \cdot 5}\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r48781177 = 1.0;
        double r48781178 = 5.0;
        double r48781179 = v;
        double r48781180 = r48781179 * r48781179;
        double r48781181 = r48781178 * r48781180;
        double r48781182 = r48781177 - r48781181;
        double r48781183 = r48781180 - r48781177;
        double r48781184 = r48781182 / r48781183;
        double r48781185 = acos(r48781184);
        return r48781185;
}
↓
double f(double v) {
        double r48781186 = 1.0;
        double r48781187 = v;
        double r48781188 = r48781187 * r48781187;
        double r48781189 = 5.0;
        double r48781190 = r48781188 * r48781189;
        double r48781191 = cbrt(r48781190);
        double r48781192 = r48781191 * r48781191;
        double r48781193 = r48781191 * r48781192;
        double r48781194 = r48781186 - r48781193;
        double r48781195 = r48781188 - r48781186;
        double r48781196 = r48781194 / r48781195;
        double r48781197 = acos(r48781196);
        return r48781197;
}

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-cube-cbrt0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019107 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))