Average Error: 0.5 → 0.5
Time: 5.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)
double f(double v) {
        double r314911 = 1.0;
        double r314912 = 5.0;
        double r314913 = v;
        double r314914 = r314913 * r314913;
        double r314915 = r314912 * r314914;
        double r314916 = r314911 - r314915;
        double r314917 = r314914 - r314911;
        double r314918 = r314916 / r314917;
        double r314919 = acos(r314918);
        return r314919;
}
↓
double f(double v) {
        double r314920 = 1.0;
        double r314921 = 3.0;
        double r314922 = pow(r314920, r314921);
        double r314923 = 5.0;
        double r314924 = v;
        double r314925 = r314924 * r314924;
        double r314926 = r314923 * r314925;
        double r314927 = pow(r314926, r314921);
        double r314928 = r314922 - r314927;
        double r314929 = r314926 + r314920;
        double r314930 = r314926 * r314929;
        double r314931 = r314920 * r314920;
        double r314932 = r314930 + r314931;
        double r314933 = r314925 - r314920;
        double r314934 = r314932 * r314933;
        double r314935 = r314928 / r314934;
        double r314936 = acos(r314935);
        return r314936;
}

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 flip3--0.5

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{1 \cdot 1 + \left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(5 \cdot \left(v \cdot v\right)\right)\right)}}}{v \cdot v - 1}\right)\]

  4. Applied associate-/l/0.5

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(v \cdot v - 1\right) \cdot \left(1 \cdot 1 + \left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(5 \cdot \left(v \cdot v\right)\right)\right)\right)}\right)}\]

  5. Simplified0.5

    \[\leadsto \cos^{-1} \left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\color{blue}{\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}}\right)\]

  6. Final simplification0.5

    \[\leadsto \cos^{-1} \left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))