Average Error: 0.5 → 0.5
Time: 15.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)
double f(double v) {
        double r217342 = 1.0;
        double r217343 = 5.0;
        double r217344 = v;
        double r217345 = r217344 * r217344;
        double r217346 = r217343 * r217345;
        double r217347 = r217342 - r217346;
        double r217348 = r217345 - r217342;
        double r217349 = r217347 / r217348;
        double r217350 = acos(r217349);
        return r217350;
}
↓
double f(double v) {
        double r217351 = atan2(1.0, 0.0);
        double r217352 = 2.0;
        double r217353 = r217351 / r217352;
        double r217354 = 1.0;
        double r217355 = v;
        double r217356 = r217355 * r217355;
        double r217357 = 1.0;
        double r217358 = r217356 - r217357;
        double r217359 = 5.0;
        double r217360 = r217359 * r217356;
        double r217361 = r217357 - r217360;
        double r217362 = r217358 / r217361;
        double r217363 = r217354 / r217362;
        double r217364 = asin(r217363);
        double r217365 = r217353 - r217364;
        return r217365;
}

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 clear-num0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020043 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))