Average Error: 0.5 → 0.5
Time: 52.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{e}^{\left(\log \left(\cos^{-1} \left(\sqrt{1 - \left(v \cdot v\right) \cdot 5} \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{v \cdot v - 1}\right)\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{e}^{\left(\log \left(\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)\right)\right)}
double f(double v) {
        double r41352212 = 1.0;
        double r41352213 = 5.0;
        double r41352214 = v;
        double r41352215 = r41352214 * r41352214;
        double r41352216 = r41352213 * r41352215;
        double r41352217 = r41352212 - r41352216;
        double r41352218 = r41352215 - r41352212;
        double r41352219 = r41352217 / r41352218;
        double r41352220 = acos(r41352219);
        return r41352220;
}
↓
double f(double v) {
        double r41352221 = exp(1.0);
        double r41352222 = 1.0;
        double r41352223 = v;
        double r41352224 = r41352223 * r41352223;
        double r41352225 = 5.0;
        double r41352226 = r41352224 * r41352225;
        double r41352227 = r41352222 - r41352226;
        double r41352228 = sqrt(r41352227);
        double r41352229 = r41352224 - r41352222;
        double r41352230 = r41352228 / r41352229;
        double r41352231 = r41352228 * r41352230;
        double r41352232 = acos(r41352231);
        double r41352233 = log(r41352232);
        double r41352234 = pow(r41352221, r41352233);
        return r41352234;
}

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 *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied times-frac0.5

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)}\]
  6. Using strategy rm

  7. Applied add-exp-log0.5

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\right)}}\]
  8. Using strategy rm

  9. Applied pow10.5

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

  10. Applied log-pow0.5

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

  11. Applied exp-prod0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019124 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))