Average Error: 0.5 → 0.6
Time: 11.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r321909 = 1.0;
        double r321910 = 5.0;
        double r321911 = v;
        double r321912 = r321911 * r321911;
        double r321913 = r321910 * r321912;
        double r321914 = r321909 - r321913;
        double r321915 = r321912 - r321909;
        double r321916 = r321914 / r321915;
        double r321917 = acos(r321916);
        return r321917;
}
↓
double f(double v) {
        double r321918 = 1.0;
        double r321919 = 5.0;
        double r321920 = v;
        double r321921 = r321920 * r321920;
        double r321922 = r321919 * r321921;
        double r321923 = r321918 - r321922;
        double r321924 = sqrt(r321923);
        double r321925 = r321921 - r321918;
        double r321926 = r321924 / r321925;
        double r321927 = r321924 * r321926;
        double r321928 = acos(r321927);
        return r321928;
}

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 *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied times-frac0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))