Average Error: 0.6 → 0.6
Time: 29.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{e}^{\left(\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{e}^{\left(\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}
double f(double v) {
        double r121996 = 1.0;
        double r121997 = 5.0;
        double r121998 = v;
        double r121999 = r121998 * r121998;
        double r122000 = r121997 * r121999;
        double r122001 = r121996 - r122000;
        double r122002 = r121999 - r121996;
        double r122003 = r122001 / r122002;
        double r122004 = acos(r122003);
        return r122004;
}
↓
double f(double v) {
        double r122005 = exp(1.0);
        double r122006 = 1.0;
        double r122007 = 5.0;
        double r122008 = v;
        double r122009 = r122008 * r122008;
        double r122010 = r122007 * r122009;
        double r122011 = r122006 - r122010;
        double r122012 = r122009 - r122006;
        double r122013 = r122011 / r122012;
        double r122014 = acos(r122013);
        double r122015 = log(r122014);
        double r122016 = pow(r122005, r122015);
        return r122016;
}

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-exp-log0.6

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

  5. Applied pow10.6

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

  6. Applied log-pow0.6

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

  7. Applied exp-prod0.6

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

  8. Simplified0.6

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

  10. Applied pow10.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))