Average Error: 0.5 → 0.7
Time: 5.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}
double f(double v) {
        double r185324 = 1.0;
        double r185325 = 5.0;
        double r185326 = v;
        double r185327 = r185326 * r185326;
        double r185328 = r185325 * r185327;
        double r185329 = r185324 - r185328;
        double r185330 = r185327 - r185324;
        double r185331 = r185329 / r185330;
        double r185332 = acos(r185331);
        return r185332;
}
↓
double f(double v) {
        double r185333 = 4.0;
        double r185334 = v;
        double r185335 = 2.0;
        double r185336 = pow(r185334, r185335);
        double r185337 = 4.0;
        double r185338 = pow(r185334, r185337);
        double r185339 = r185336 + r185338;
        double r185340 = r185333 * r185339;
        double r185341 = 1.0;
        double r185342 = r185340 - r185341;
        double r185343 = acos(r185342);
        double r185344 = log(r185343);
        double r185345 = exp(r185344);
        return r185345;
}

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. Taylor expanded around 0 0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\]

  3. Simplified0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]
  4. Using strategy rm

  5. Applied add-exp-log0.7

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}}\]

  6. Final simplification0.7

    \[\leadsto e^{\log \left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))