Average Error: 0.5 → 0.6
Time: 25.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{\frac{v \cdot v - 1}{v \cdot v}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{\frac{v \cdot v - 1}{v \cdot v}}\right)
double f(double v) {
        double r12783869 = 1.0;
        double r12783870 = 5.0;
        double r12783871 = v;
        double r12783872 = r12783871 * r12783871;
        double r12783873 = r12783870 * r12783872;
        double r12783874 = r12783869 - r12783873;
        double r12783875 = r12783872 - r12783869;
        double r12783876 = r12783874 / r12783875;
        double r12783877 = acos(r12783876);
        return r12783877;
}
↓
double f(double v) {
        double r12783878 = 1.0;
        double r12783879 = v;
        double r12783880 = r12783879 * r12783879;
        double r12783881 = r12783880 - r12783878;
        double r12783882 = r12783878 / r12783881;
        double r12783883 = 5.0;
        double r12783884 = r12783881 / r12783880;
        double r12783885 = r12783883 / r12783884;
        double r12783886 = r12783882 - r12783885;
        double r12783887 = acos(r12783886);
        return r12783887;
}

Error

Bits error versus v

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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 expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))