Average Error: 0.5 → 0.5
Time: 22.1s
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 r5983044 = 1.0;
        double r5983045 = 5.0;
        double r5983046 = v;
        double r5983047 = r5983046 * r5983046;
        double r5983048 = r5983045 * r5983047;
        double r5983049 = r5983044 - r5983048;
        double r5983050 = r5983047 - r5983044;
        double r5983051 = r5983049 / r5983050;
        double r5983052 = acos(r5983051);
        return r5983052;
}
↓
double f(double v) {
        double r5983053 = 1.0;
        double r5983054 = v;
        double r5983055 = r5983054 * r5983054;
        double r5983056 = r5983055 - r5983053;
        double r5983057 = r5983053 / r5983056;
        double r5983058 = 5.0;
        double r5983059 = r5983056 / r5983055;
        double r5983060 = r5983058 / r5983059;
        double r5983061 = r5983057 - r5983060;
        double r5983062 = acos(r5983061);
        double r5983063 = r5983055 * r5983058;
        double r5983064 = r5983053 - r5983063;
        double r5983065 = r5983064 / r5983056;
        double r5983066 = acos(r5983065);
        double r5983067 = r5983062 * r5983066;
        double r5983068 = sqrt(r5983067);
        return r5983068;
}

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))))