Average Error: 0.6 → 0.8
Time: 24.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\left(4 \cdot \left(v \cdot v\right) + \left(4 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right) - 1\right)\]
\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) + \left(4 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right) - 1\right)
double f(double v) {
        double r7381649 = 1.0;
        double r7381650 = 5.0;
        double r7381651 = v;
        double r7381652 = r7381651 * r7381651;
        double r7381653 = r7381650 * r7381652;
        double r7381654 = r7381649 - r7381653;
        double r7381655 = r7381652 - r7381649;
        double r7381656 = r7381654 / r7381655;
        double r7381657 = acos(r7381656);
        return r7381657;
}
↓
double f(double v) {
        double r7381658 = 4.0;
        double r7381659 = v;
        double r7381660 = r7381659 * r7381659;
        double r7381661 = r7381658 * r7381660;
        double r7381662 = r7381661 * r7381660;
        double r7381663 = r7381661 + r7381662;
        double r7381664 = 1.0;
        double r7381665 = r7381663 - r7381664;
        double r7381666 = acos(r7381665);
        return r7381666;
}

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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019163 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))