Average Error: 0.5 → 0.5
Time: 24.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\left(\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\left(\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}} \cdot \sqrt[3]{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}\right)
double f(double v) {
        double r12472295 = 1.0;
        double r12472296 = 5.0;
        double r12472297 = v;
        double r12472298 = r12472297 * r12472297;
        double r12472299 = r12472296 * r12472298;
        double r12472300 = r12472295 - r12472299;
        double r12472301 = r12472298 - r12472295;
        double r12472302 = r12472300 / r12472301;
        double r12472303 = acos(r12472302);
        return r12472303;
}
↓
double f(double v) {
        double r12472304 = 1.0;
        double r12472305 = v;
        double r12472306 = r12472305 * r12472305;
        double r12472307 = 5.0;
        double r12472308 = r12472306 * r12472307;
        double r12472309 = r12472304 - r12472308;
        double r12472310 = r12472306 - r12472304;
        double r12472311 = r12472309 / r12472310;
        double r12472312 = acos(r12472311);
        double r12472313 = cbrt(r12472312);
        double r12472314 = cbrt(r12472313);
        double r12472315 = r12472314 * r12472314;
        double r12472316 = r12472315 * r12472315;
        double r12472317 = r12472315 * r12472316;
        double r12472318 = sqrt(r12472312);
        double r12472319 = cbrt(r12472318);
        double r12472320 = r12472319 * r12472319;
        double r12472321 = r12472317 * r12472320;
        return r12472321;
}

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-cbrt2.0

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

  5. Applied add-cube-cbrt1.5

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

  6. Applied add-cube-cbrt1.5

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

  7. Applied swap-sqr2.0

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

  9. Applied add-sqr-sqrt2.4

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

  10. Applied cbrt-prod0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))