Average Error: 0.6 → 0.6
Time: 4.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\log \left(e^{1 - 5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\log \left(e^{1 - 5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r301919 = 1.0;
        double r301920 = 5.0;
        double r301921 = v;
        double r301922 = r301921 * r301921;
        double r301923 = r301920 * r301922;
        double r301924 = r301919 - r301923;
        double r301925 = r301922 - r301919;
        double r301926 = r301924 / r301925;
        double r301927 = acos(r301926);
        return r301927;
}
↓
double f(double v) {
        double r301928 = 1.0;
        double r301929 = 5.0;
        double r301930 = v;
        double r301931 = r301930 * r301930;
        double r301932 = r301929 * r301931;
        double r301933 = r301928 - r301932;
        double r301934 = exp(r301933);
        double r301935 = log(r301934);
        double r301936 = r301931 - r301928;
        double r301937 = r301935 / r301936;
        double r301938 = acos(r301937);
        return r301938;
}

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 add-log-exp0.6

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

  4. Applied add-log-exp0.6

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

  5. Applied diff-log0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))