Average Error: 0.6 → 0.8
Time: 23.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)
double f(double v) {
        double r166846 = 1.0;
        double r166847 = 5.0;
        double r166848 = v;
        double r166849 = r166848 * r166848;
        double r166850 = r166847 * r166849;
        double r166851 = r166846 - r166850;
        double r166852 = r166849 - r166846;
        double r166853 = r166851 / r166852;
        double r166854 = acos(r166853);
        return r166854;
}
↓
double f(double v) {
        double r166855 = v;
        double r166856 = r166855 * r166855;
        double r166857 = 4.0;
        double r166858 = pow(r166855, r166857);
        double r166859 = r166856 + r166858;
        double r166860 = 4.0;
        double r166861 = r166859 * r166860;
        double r166862 = 1.0;
        double r166863 = r166861 - r166862;
        double r166864 = acos(r166863);
        return r166864;
}

Error

Bits error versus v

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.6

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

  2. Simplified0.6

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

  3. Taylor expanded around 0 0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]

  4. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)}\]

  5. Final simplification0.8

    \[\leadsto \cos^{-1} \left(\left(v \cdot v + {v}^{4}\right) \cdot 4 - 1\right)\]

Reproduce

herbie shell --seed 2019174 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))