Average Error: 0.6 → 0.8
Time: 24.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\left(4 \cdot \left(v \cdot v\right) + \left(4 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right) - 1\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\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)\right)}
double f(double v) {
        double r5729854 = 1.0;
        double r5729855 = 5.0;
        double r5729856 = v;
        double r5729857 = r5729856 * r5729856;
        double r5729858 = r5729855 * r5729857;
        double r5729859 = r5729854 - r5729858;
        double r5729860 = r5729857 - r5729854;
        double r5729861 = r5729859 / r5729860;
        double r5729862 = acos(r5729861);
        return r5729862;
}
↓
double f(double v) {
        double r5729863 = 4.0;
        double r5729864 = v;
        double r5729865 = r5729864 * r5729864;
        double r5729866 = r5729863 * r5729865;
        double r5729867 = r5729866 * r5729865;
        double r5729868 = r5729866 + r5729867;
        double r5729869 = 1.0;
        double r5729870 = r5729868 - r5729869;
        double r5729871 = acos(r5729870);
        double r5729872 = log(r5729871);
        double r5729873 = exp(r5729872);
        return r5729873;
}

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. Using strategy rm

  5. Applied add-exp-log0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019158 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))