Average Error: 0.6 → 0.6
Time: 7.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r358877 = 1.0;
        double r358878 = 5.0;
        double r358879 = v;
        double r358880 = r358879 * r358879;
        double r358881 = r358878 * r358880;
        double r358882 = r358877 - r358881;
        double r358883 = r358880 - r358877;
        double r358884 = r358882 / r358883;
        double r358885 = acos(r358884);
        return r358885;
}
↓
double f(double v) {
        double r358886 = atan2(1.0, 0.0);
        double r358887 = 2.0;
        double r358888 = r358886 / r358887;
        double r358889 = 1.0;
        double r358890 = 5.0;
        double r358891 = v;
        double r358892 = r358891 * r358891;
        double r358893 = r358890 * r358892;
        double r358894 = r358889 - r358893;
        double r358895 = r358892 - r358889;
        double r358896 = r358894 / r358895;
        double r358897 = asin(r358896);
        double r358898 = r358888 - r358897;
        return r358898;
}

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. Using strategy rm

  3. Applied acos-asin0.6

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]

  4. Final simplification0.6

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))