Average Error: 0.6 → 0.8
Time: 19.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) \cdot 4 - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) \cdot 4 - 1\right)
double f(double v) {
        double r7741556 = 1.0;
        double r7741557 = 5.0;
        double r7741558 = v;
        double r7741559 = r7741558 * r7741558;
        double r7741560 = r7741557 * r7741559;
        double r7741561 = r7741556 - r7741560;
        double r7741562 = r7741559 - r7741556;
        double r7741563 = r7741561 / r7741562;
        double r7741564 = acos(r7741563);
        return r7741564;
}
↓
double f(double v) {
        double r7741565 = v;
        double r7741566 = r7741565 * r7741565;
        double r7741567 = r7741566 * r7741566;
        double r7741568 = r7741567 + r7741566;
        double r7741569 = 4.0;
        double r7741570 = r7741568 * r7741569;
        double r7741571 = 1.0;
        double r7741572 = r7741570 - r7741571;
        double r7741573 = acos(r7741572);
        return r7741573;
}

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 0 0.8

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

  3. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) - 1\right)}\]

  4. Final simplification0.8

    \[\leadsto \cos^{-1} \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + v \cdot v\right) \cdot 4 - 1\right)\]

Reproduce

herbie shell --seed 2019174 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))