Average Error: 0.6 → 0.7
Time: 4.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\left(\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\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)}}{v \cdot v - 1}\right)
double f(double v) {
        double r328999 = 1.0;
        double r329000 = 5.0;
        double r329001 = v;
        double r329002 = r329001 * r329001;
        double r329003 = r329000 * r329002;
        double r329004 = r328999 - r329003;
        double r329005 = r329002 - r328999;
        double r329006 = r329004 / r329005;
        double r329007 = acos(r329006);
        return r329007;
}
↓
double f(double v) {
        double r329008 = 1.0;
        double r329009 = 5.0;
        double r329010 = v;
        double r329011 = r329010 * r329010;
        double r329012 = r329009 * r329011;
        double r329013 = r329008 - r329012;
        double r329014 = cbrt(r329013);
        double r329015 = r329014 * r329014;
        double r329016 = r329015 * r329014;
        double r329017 = r329011 - r329008;
        double r329018 = r329016 / r329017;
        double r329019 = acos(r329018);
        return r329019;
}

Error

Bits error versus v

Try it out

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-cube-cbrt0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))