Average Error: 0.5 → 0.7
Time: 18.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\left(4 \cdot \left(v \cdot v\right) - 1\right) + \left(v \cdot v\right) \cdot \left(4 \cdot \left(v \cdot v\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\left(4 \cdot \left(v \cdot v\right) - 1\right) + \left(v \cdot v\right) \cdot \left(4 \cdot \left(v \cdot v\right)\right)\right)
double f(double v) {
        double r3208750 = 1.0;
        double r3208751 = 5.0;
        double r3208752 = v;
        double r3208753 = r3208752 * r3208752;
        double r3208754 = r3208751 * r3208753;
        double r3208755 = r3208750 - r3208754;
        double r3208756 = r3208753 - r3208750;
        double r3208757 = r3208755 / r3208756;
        double r3208758 = acos(r3208757);
        return r3208758;
}
↓
double f(double v) {
        double r3208759 = 4.0;
        double r3208760 = v;
        double r3208761 = r3208760 * r3208760;
        double r3208762 = r3208759 * r3208761;
        double r3208763 = 1.0;
        double r3208764 = r3208762 - r3208763;
        double r3208765 = r3208761 * r3208762;
        double r3208766 = r3208764 + r3208765;
        double r3208767 = acos(r3208766);
        return r3208767;
}

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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))