Average Error: 0.6 → 0.6
Time: 21.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt{1 - \left(v \cdot v\right) \cdot 5} \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt{1 - \left(v \cdot v\right) \cdot 5} \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{v \cdot v - 1}\right)
double f(double v) {
        double r7786725 = 1.0;
        double r7786726 = 5.0;
        double r7786727 = v;
        double r7786728 = r7786727 * r7786727;
        double r7786729 = r7786726 * r7786728;
        double r7786730 = r7786725 - r7786729;
        double r7786731 = r7786728 - r7786725;
        double r7786732 = r7786730 / r7786731;
        double r7786733 = acos(r7786732);
        return r7786733;
}
↓
double f(double v) {
        double r7786734 = 1.0;
        double r7786735 = v;
        double r7786736 = r7786735 * r7786735;
        double r7786737 = 5.0;
        double r7786738 = r7786736 * r7786737;
        double r7786739 = r7786734 - r7786738;
        double r7786740 = sqrt(r7786739);
        double r7786741 = r7786736 - r7786734;
        double r7786742 = r7786740 / r7786741;
        double r7786743 = r7786740 * r7786742;
        double r7786744 = acos(r7786743);
        return r7786744;
}

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 *-un-lft-identity0.6

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

  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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))