Average Error: 0.5 → 0.6
Time: 22.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}
double f(double v) {
        double r201498 = 1.0;
        double r201499 = 5.0;
        double r201500 = v;
        double r201501 = r201500 * r201500;
        double r201502 = r201499 * r201501;
        double r201503 = r201498 - r201502;
        double r201504 = r201501 - r201498;
        double r201505 = r201503 / r201504;
        double r201506 = acos(r201505);
        return r201506;
}
↓
double f(double v) {
        double r201507 = 1.0;
        double r201508 = v;
        double r201509 = r201508 * r201508;
        double r201510 = 5.0;
        double r201511 = r201509 * r201510;
        double r201512 = r201507 - r201511;
        double r201513 = r201509 - r201507;
        double r201514 = r201512 / r201513;
        double r201515 = acos(r201514);
        double r201516 = sqrt(r201515);
        double r201517 = sqrt(r201516);
        double r201518 = cbrt(r201517);
        double r201519 = r201518 * r201518;
        double r201520 = 3.0;
        double r201521 = pow(r201519, r201520);
        double r201522 = 2.0;
        double r201523 = pow(r201508, r201522);
        double r201524 = r201510 * r201523;
        double r201525 = r201507 - r201524;
        double r201526 = r201523 - r201507;
        double r201527 = r201525 / r201526;
        double r201528 = acos(r201527);
        double r201529 = sqrt(r201528);
        double r201530 = r201521 * r201529;
        return r201530;
}

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-sqr-sqrt1.5

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

  4. Simplified1.5

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

  5. Simplified1.5

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

  7. Applied add-sqr-sqrt1.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied associate-*r*1.5

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

  10. Simplified1.5

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

  12. Applied add-cube-cbrt1.5

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

  13. Applied unpow-prod-down2.1

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

  14. Applied associate-*l*2.1

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

  15. Simplified0.6

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

  16. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))