Average Error: 0.5 → 0.6
Time: 5.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)
double f(double v) {
        double r230679 = 1.0;
        double r230680 = 5.0;
        double r230681 = v;
        double r230682 = r230681 * r230681;
        double r230683 = r230680 * r230682;
        double r230684 = r230679 - r230683;
        double r230685 = r230682 - r230679;
        double r230686 = r230684 / r230685;
        double r230687 = acos(r230686);
        return r230687;
}
↓
double f(double v) {
        double r230688 = 1.0;
        double r230689 = 5.0;
        double r230690 = v;
        double r230691 = r230690 * r230690;
        double r230692 = r230689 * r230691;
        double r230693 = r230688 - r230692;
        double r230694 = cbrt(r230693);
        double r230695 = r230694 * r230694;
        double r230696 = r230691 - r230688;
        double r230697 = cbrt(r230696);
        double r230698 = r230697 * r230697;
        double r230699 = r230695 / r230698;
        double r230700 = r230694 / r230697;
        double r230701 = r230699 * r230700;
        double r230702 = acos(r230701);
        return r230702;
}

Error

Bits error versus v

Try it out

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 add-cube-cbrt0.6

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

  4. Applied add-cube-cbrt0.6

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

  5. Applied times-frac0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2020060 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))