Average Error: 0.6 → 0.7
Time: 14.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \mathsf{fma}\left(v, v, {v}^{4}\right) - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \mathsf{fma}\left(v, v, {v}^{4}\right) - 1\right)
double f(double v) {
        double r191156 = 1.0;
        double r191157 = 5.0;
        double r191158 = v;
        double r191159 = r191158 * r191158;
        double r191160 = r191157 * r191159;
        double r191161 = r191156 - r191160;
        double r191162 = r191159 - r191156;
        double r191163 = r191161 / r191162;
        double r191164 = acos(r191163);
        return r191164;
}
↓
double f(double v) {
        double r191165 = atan2(1.0, 0.0);
        double r191166 = 2.0;
        double r191167 = r191165 / r191166;
        double r191168 = 4.0;
        double r191169 = v;
        double r191170 = 4.0;
        double r191171 = pow(r191169, r191170);
        double r191172 = fma(r191169, r191169, r191171);
        double r191173 = r191168 * r191172;
        double r191174 = 1.0;
        double r191175 = r191173 - r191174;
        double r191176 = asin(r191175);
        double r191177 = r191167 - r191176;
        return r191177;
}

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 0.7

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

  3. Simplified0.7

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

  5. Applied acos-asin0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))