Average Error: 0.6 → 0.6
Time: 20.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - 1 \cdot 1} \cdot \left(v \cdot v + 1\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - 1 \cdot 1} \cdot \left(v \cdot v + 1\right)\right)
double f(double v) {
        double r9386942 = 1.0;
        double r9386943 = 5.0;
        double r9386944 = v;
        double r9386945 = r9386944 * r9386944;
        double r9386946 = r9386943 * r9386945;
        double r9386947 = r9386942 - r9386946;
        double r9386948 = r9386945 - r9386942;
        double r9386949 = r9386947 / r9386948;
        double r9386950 = acos(r9386949);
        return r9386950;
}
↓
double f(double v) {
        double r9386951 = 1.0;
        double r9386952 = v;
        double r9386953 = r9386952 * r9386952;
        double r9386954 = 5.0;
        double r9386955 = r9386953 * r9386954;
        double r9386956 = r9386951 - r9386955;
        double r9386957 = r9386953 * r9386953;
        double r9386958 = r9386951 * r9386951;
        double r9386959 = r9386957 - r9386958;
        double r9386960 = r9386956 / r9386959;
        double r9386961 = r9386953 + r9386951;
        double r9386962 = r9386960 * r9386961;
        double r9386963 = acos(r9386962);
        return r9386963;
}

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 flip--0.6

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

  4. Applied associate-/r/0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019192 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))