Average Error: 0.6 → 0.8
Time: 5.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)
double f(double v) {
        double r127463 = 1.0;
        double r127464 = 5.0;
        double r127465 = v;
        double r127466 = r127465 * r127465;
        double r127467 = r127464 * r127466;
        double r127468 = r127463 - r127467;
        double r127469 = r127466 - r127463;
        double r127470 = r127468 / r127469;
        double r127471 = acos(r127470);
        return r127471;
}
↓
double f(double v) {
        double r127472 = atan2(1.0, 0.0);
        double r127473 = 2.0;
        double r127474 = r127472 / r127473;
        double r127475 = 4.0;
        double r127476 = v;
        double r127477 = 4.0;
        double r127478 = pow(r127476, r127477);
        double r127479 = fma(r127476, r127476, r127478);
        double r127480 = 1.0;
        double r127481 = -r127480;
        double r127482 = fma(r127475, r127479, r127481);
        double r127483 = asin(r127482);
        double r127484 = r127474 - r127483;
        return r127484;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)}\]
  4. Using strategy rm

  5. Applied acos-asin0.8

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)}\]

  6. Final simplification0.8

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))