Average Error: 0.5 → 0.7
Time: 39.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sqrt[3]{\sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right) \cdot \sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sqrt[3]{\sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right) \cdot \sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right)\right)}
double f(double v) {
        double r8635460 = 1.0;
        double r8635461 = 5.0;
        double r8635462 = v;
        double r8635463 = r8635462 * r8635462;
        double r8635464 = r8635461 * r8635463;
        double r8635465 = r8635460 - r8635464;
        double r8635466 = r8635463 - r8635460;
        double r8635467 = r8635465 / r8635466;
        double r8635468 = acos(r8635467);
        return r8635468;
}
↓
double f(double v) {
        double r8635469 = atan2(1.0, 0.0);
        double r8635470 = 2.0;
        double r8635471 = r8635469 / r8635470;
        double r8635472 = 4.0;
        double r8635473 = v;
        double r8635474 = r8635473 * r8635473;
        double r8635475 = r8635474 * r8635474;
        double r8635476 = r8635475 + r8635474;
        double r8635477 = r8635472 * r8635476;
        double r8635478 = 1.0;
        double r8635479 = r8635477 - r8635478;
        double r8635480 = asin(r8635479);
        double r8635481 = r8635480 * r8635480;
        double r8635482 = r8635480 * r8635481;
        double r8635483 = cbrt(r8635482);
        double r8635484 = r8635471 - r8635483;
        return r8635484;
}

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 acos-asin0.5

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

  4. Taylor expanded around 0 0.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]

  5. Simplified0.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 4 - 1\right)}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube0.7

    \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt[3]{\left(\sin^{-1} \left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 4 - 1\right) \cdot \sin^{-1} \left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 4 - 1\right)\right) \cdot \sin^{-1} \left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 4 - 1\right)}}\]

  8. Final simplification0.7

    \[\leadsto \frac{\pi}{2} - \sqrt[3]{\sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right) \cdot \left(\sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right) \cdot \sin^{-1} \left(4 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) - 1\right)\right)}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))