Average Error: 0.5 → 0.7
Time: 5.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)
double f(double v) {
        double r216676 = 1.0;
        double r216677 = 5.0;
        double r216678 = v;
        double r216679 = r216678 * r216678;
        double r216680 = r216677 * r216679;
        double r216681 = r216676 - r216680;
        double r216682 = r216679 - r216676;
        double r216683 = r216681 / r216682;
        double r216684 = acos(r216683);
        return r216684;
}
↓
double f(double v) {
        double r216685 = 4.0;
        double r216686 = v;
        double r216687 = 2.0;
        double r216688 = pow(r216686, r216687);
        double r216689 = 4.0;
        double r216690 = pow(r216686, r216689);
        double r216691 = r216688 + r216690;
        double r216692 = r216685 * r216691;
        double r216693 = 1.0;
        double r216694 = r216692 - r216693;
        double r216695 = acos(r216694);
        return r216695;
}

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

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020060 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))