Average Error: 0.6 → 0.6
Time: 30.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}^{\frac{1}{3}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}^{\frac{1}{3}}
double f(double v) {
        double r4895955 = 1.0;
        double r4895956 = 5.0;
        double r4895957 = v;
        double r4895958 = r4895957 * r4895957;
        double r4895959 = r4895956 * r4895958;
        double r4895960 = r4895955 - r4895959;
        double r4895961 = r4895958 - r4895955;
        double r4895962 = r4895960 / r4895961;
        double r4895963 = acos(r4895962);
        return r4895963;
}
↓
double f(double v) {
        double r4895964 = 1.0;
        double r4895965 = v;
        double r4895966 = r4895965 * r4895965;
        double r4895967 = 5.0;
        double r4895968 = r4895966 * r4895967;
        double r4895969 = r4895964 - r4895968;
        double r4895970 = r4895966 - r4895964;
        double r4895971 = r4895969 / r4895970;
        double r4895972 = acos(r4895971);
        double r4895973 = r4895972 * r4895972;
        double r4895974 = r4895973 * r4895972;
        double r4895975 = 0.3333333333333333;
        double r4895976 = pow(r4895974, r4895975);
        return r4895976;
}

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube1.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\]
  4. Using strategy rm

  5. Applied pow1/30.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019144 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))