Average Error: 0.5 → 0.5
Time: 23.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt{\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{\frac{v \cdot v - 1}{v \cdot v}}\right) \cdot \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt{\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{\frac{v \cdot v - 1}{v \cdot v}}\right) \cdot \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}
double f(double v) {
        double r5983042 = 1.0;
        double r5983043 = 5.0;
        double r5983044 = v;
        double r5983045 = r5983044 * r5983044;
        double r5983046 = r5983043 * r5983045;
        double r5983047 = r5983042 - r5983046;
        double r5983048 = r5983045 - r5983042;
        double r5983049 = r5983047 / r5983048;
        double r5983050 = acos(r5983049);
        return r5983050;
}
↓
double f(double v) {
        double r5983051 = 1.0;
        double r5983052 = v;
        double r5983053 = r5983052 * r5983052;
        double r5983054 = r5983053 - r5983051;
        double r5983055 = r5983051 / r5983054;
        double r5983056 = 5.0;
        double r5983057 = r5983054 / r5983053;
        double r5983058 = r5983056 / r5983057;
        double r5983059 = r5983055 - r5983058;
        double r5983060 = acos(r5983059);
        double r5983061 = r5983053 * r5983056;
        double r5983062 = r5983051 - r5983061;
        double r5983063 = r5983062 / r5983054;
        double r5983064 = acos(r5983063);
        double r5983065 = r5983060 * r5983064;
        double r5983066 = sqrt(r5983065);
        return r5983066;
}

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. Using strategy rm

  5. Applied add-sqr-sqrt1.5

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

  6. Taylor expanded around 0 1.5

    \[\leadsto \sqrt{\cos^{-1} \left(\sqrt[3]{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right)} \cdot \color{blue}{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}}\]

  7. Simplified1.5

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

  9. Applied sqrt-unprod0.6

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

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))))