Average Error: 0.5 → 0.5
Time: 5.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}\right)\right)
double f(double v) {
        double r168117 = 1.0;
        double r168118 = 5.0;
        double r168119 = v;
        double r168120 = r168119 * r168119;
        double r168121 = r168118 * r168120;
        double r168122 = r168117 - r168121;
        double r168123 = r168120 - r168117;
        double r168124 = r168122 / r168123;
        double r168125 = acos(r168124);
        return r168125;
}
↓
double f(double v) {
        double r168126 = atan2(1.0, 0.0);
        double r168127 = 2.0;
        double r168128 = r168126 / r168127;
        double r168129 = 1.0;
        double r168130 = 5.0;
        double r168131 = v;
        double r168132 = r168131 * r168131;
        double r168133 = r168130 * r168132;
        double r168134 = r168129 - r168133;
        double r168135 = r168132 - r168129;
        double r168136 = r168134 / r168135;
        double r168137 = asin(r168136);
        double r168138 = r168128 - r168137;
        double r168139 = 3.0;
        double r168140 = pow(r168138, r168139);
        double r168141 = cbrt(r168140);
        double r168142 = log1p(r168141);
        double r168143 = expm1(r168142);
        return r168143;
}

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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube1.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right) \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]

  6. Simplified1.5

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}}\]
  7. Using strategy rm

  8. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}\right)\right)}\]

  9. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}\right)\right)\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))