Average Error: 0.5 → 0.6
Time: 7.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\frac{v \cdot v - 1}{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\frac{v \cdot v - 1}{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}}\right)
double f(double v) {
        double r381210 = 1.0;
        double r381211 = 5.0;
        double r381212 = v;
        double r381213 = r381212 * r381212;
        double r381214 = r381211 * r381213;
        double r381215 = r381210 - r381214;
        double r381216 = r381213 - r381210;
        double r381217 = r381215 / r381216;
        double r381218 = acos(r381217);
        return r381218;
}
↓
double f(double v) {
        double r381219 = atan2(1.0, 0.0);
        double r381220 = 2.0;
        double r381221 = r381219 / r381220;
        double r381222 = 1.0;
        double r381223 = 5.0;
        double r381224 = v;
        double r381225 = r381224 * r381224;
        double r381226 = r381223 * r381225;
        double r381227 = r381222 - r381226;
        double r381228 = cbrt(r381227);
        double r381229 = r381228 * r381228;
        double r381230 = r381225 - r381222;
        double r381231 = r381230 / r381228;
        double r381232 = r381229 / r381231;
        double r381233 = asin(r381232);
        double r381234 = r381221 - r381233;
        return r381234;
}

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-cube-cbrt0.6

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

  6. Applied associate-/l*0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))