Average Error: 0.5 → 0.8
Time: 25.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)\right)
double f(double v) {
        double r258354 = 1.0;
        double r258355 = 5.0;
        double r258356 = v;
        double r258357 = r258356 * r258356;
        double r258358 = r258355 * r258357;
        double r258359 = r258354 - r258358;
        double r258360 = r258357 - r258354;
        double r258361 = r258359 / r258360;
        double r258362 = acos(r258361);
        return r258362;
}
↓
double f(double v) {
        double r258363 = 4.0;
        double r258364 = v;
        double r258365 = 4.0;
        double r258366 = pow(r258364, r258365);
        double r258367 = fma(r258364, r258364, r258366);
        double r258368 = 1.0;
        double r258369 = -r258368;
        double r258370 = fma(r258363, r258367, r258369);
        double r258371 = acos(r258370);
        double r258372 = log1p(r258371);
        double r258373 = expm1(r258372);
        return r258373;
}

Error

Bits error versus v

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 expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]

  4. Taylor expanded around 0 0.8

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\right)\right)\]

  5. Simplified0.8

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)}\right)\right)\]

  6. Final simplification0.8

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))