Average Error: 0.5 → 0.5
Time: 8.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
\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}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r222838 = 1.0;
        double r222839 = 5.0;
        double r222840 = v;
        double r222841 = r222840 * r222840;
        double r222842 = r222839 * r222841;
        double r222843 = r222838 - r222842;
        double r222844 = r222841 - r222838;
        double r222845 = r222843 / r222844;
        double r222846 = acos(r222845);
        return r222846;
}
↓
double f(double v) {
        double r222847 = atan2(1.0, 0.0);
        double r222848 = 2.0;
        double r222849 = r222847 / r222848;
        double r222850 = 1.0;
        double r222851 = v;
        double r222852 = pow(r222851, r222848);
        double r222853 = r222852 - r222850;
        double r222854 = r222850 / r222853;
        double r222855 = 5.0;
        double r222856 = r222851 * r222851;
        double r222857 = r222855 * r222856;
        double r222858 = r222856 - r222850;
        double r222859 = r222857 / r222858;
        double r222860 = r222854 - r222859;
        double r222861 = asin(r222860);
        double r222862 = r222849 - r222861;
        return r222862;
}

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. Using strategy rm

  3. Applied div-sub0.5

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

  4. Simplified0.5

    \[\leadsto \cos^{-1} \left(\color{blue}{\frac{1}{{v}^{2} - 1}} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  5. Using strategy rm

  6. Applied acos-asin0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))