Average Error: 0.5 → 0.7
Time: 23.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)}}
double f(double v) {
        double r5730317 = 1.0;
        double r5730318 = 5.0;
        double r5730319 = v;
        double r5730320 = r5730319 * r5730319;
        double r5730321 = r5730318 * r5730320;
        double r5730322 = r5730317 - r5730321;
        double r5730323 = r5730320 - r5730317;
        double r5730324 = r5730322 / r5730323;
        double r5730325 = acos(r5730324);
        return r5730325;
}
↓
double f(double v) {
        double r5730326 = v;
        double r5730327 = 4.0;
        double r5730328 = pow(r5730326, r5730327);
        double r5730329 = fma(r5730326, r5730326, r5730328);
        double r5730330 = r5730329 * r5730327;
        double r5730331 = 1.0;
        double r5730332 = r5730330 - r5730331;
        double r5730333 = acos(r5730332);
        double r5730334 = log(r5730333);
        double r5730335 = sqrt(r5730334);
        double r5730336 = r5730335 * r5730335;
        double r5730337 = exp(r5730336);
        return r5730337;
}

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. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied add-exp-log0.7

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

  7. Applied add-sqr-sqrt0.7

    \[\leadsto e^{\color{blue}{\sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)}}}\]

  8. Final simplification0.7

    \[\leadsto e^{\sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\mathsf{fma}\left(v, v, {v}^{4}\right) \cdot 4 - 1\right)\right)}}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))