Average Error: 0.5 → 0.5
Time: 16.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt{\cos^{-1} \left(\frac{1}{-1 + v \cdot v} + \frac{-5}{\frac{-1 + v \cdot v}{v \cdot v}}\right)} \cdot \left(\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \cdot \sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt{\cos^{-1} \left(\frac{1}{-1 + v \cdot v} + \frac{-5}{\frac{-1 + v \cdot v}{v \cdot v}}\right)} \cdot \left(\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \cdot \sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\right)
double f(double v) {
        double r2698501 = 1.0;
        double r2698502 = 5.0;
        double r2698503 = v;
        double r2698504 = r2698503 * r2698503;
        double r2698505 = r2698502 * r2698504;
        double r2698506 = r2698501 - r2698505;
        double r2698507 = r2698504 - r2698501;
        double r2698508 = r2698506 / r2698507;
        double r2698509 = acos(r2698508);
        return r2698509;
}
↓
double f(double v) {
        double r2698510 = 1.0;
        double r2698511 = -1.0;
        double r2698512 = v;
        double r2698513 = r2698512 * r2698512;
        double r2698514 = r2698511 + r2698513;
        double r2698515 = r2698510 / r2698514;
        double r2698516 = -5.0;
        double r2698517 = r2698514 / r2698513;
        double r2698518 = r2698516 / r2698517;
        double r2698519 = r2698515 + r2698518;
        double r2698520 = acos(r2698519);
        double r2698521 = sqrt(r2698520);
        double r2698522 = 5.0;
        double r2698523 = r2698522 * r2698513;
        double r2698524 = r2698510 - r2698523;
        double r2698525 = r2698513 - r2698510;
        double r2698526 = r2698524 / r2698525;
        double r2698527 = acos(r2698526);
        double r2698528 = sqrt(r2698527);
        double r2698529 = sqrt(r2698528);
        double r2698530 = r2698529 * r2698529;
        double r2698531 = r2698521 * r2698530;
        return r2698531;
}

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. Taylor expanded around 0 1.5

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

  5. Simplified1.5

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

  7. Applied add-sqr-sqrt1.5

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

  8. Applied sqrt-prod0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019156 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))