Average Error: 0.5 → 0.7
Time: 22.3s
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 r5159431 = 1.0;
        double r5159432 = 5.0;
        double r5159433 = v;
        double r5159434 = r5159433 * r5159433;
        double r5159435 = r5159432 * r5159434;
        double r5159436 = r5159431 - r5159435;
        double r5159437 = r5159434 - r5159431;
        double r5159438 = r5159436 / r5159437;
        double r5159439 = acos(r5159438);
        return r5159439;
}
↓
double f(double v) {
        double r5159440 = v;
        double r5159441 = r5159440 * r5159440;
        double r5159442 = r5159441 * r5159441;
        double r5159443 = r5159442 + r5159441;
        double r5159444 = 4.0;
        double r5159445 = r5159443 * r5159444;
        double r5159446 = -1.0;
        double r5159447 = r5159445 + r5159446;
        double r5159448 = acos(r5159447);
        return r5159448;
}

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. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))