Average Error: 0.6 → 0.6
Time: 22.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(\left(\cos^{-1} \left(\frac{1 - \frac{v \cdot v}{\frac{1}{5}}}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - \frac{v \cdot v}{\frac{1}{5}}}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - \frac{v \cdot v}{\frac{1}{5}}}{v \cdot v - 1}\right)\right)}^{\frac{1}{3}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(\left(\cos^{-1} \left(\frac{1 - \frac{v \cdot v}{\frac{1}{5}}}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - \frac{v \cdot v}{\frac{1}{5}}}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - \frac{v \cdot v}{\frac{1}{5}}}{v \cdot v - 1}\right)\right)}^{\frac{1}{3}}
double f(double v) {
        double r6799409 = 1.0;
        double r6799410 = 5.0;
        double r6799411 = v;
        double r6799412 = r6799411 * r6799411;
        double r6799413 = r6799410 * r6799412;
        double r6799414 = r6799409 - r6799413;
        double r6799415 = r6799412 - r6799409;
        double r6799416 = r6799414 / r6799415;
        double r6799417 = acos(r6799416);
        return r6799417;
}
↓
double f(double v) {
        double r6799418 = 1.0;
        double r6799419 = v;
        double r6799420 = r6799419 * r6799419;
        double r6799421 = 0.2;
        double r6799422 = r6799420 / r6799421;
        double r6799423 = r6799418 - r6799422;
        double r6799424 = r6799420 - r6799418;
        double r6799425 = r6799423 / r6799424;
        double r6799426 = acos(r6799425);
        double r6799427 = r6799426 * r6799426;
        double r6799428 = r6799427 * r6799426;
        double r6799429 = 0.3333333333333333;
        double r6799430 = pow(r6799428, r6799429);
        return r6799430;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around inf 0.6

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

  3. Simplified0.6

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

  5. Applied add-cbrt-cube1.5

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

  7. Applied pow1/30.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019130 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))