Average Error: 0.6 → 0.6
Time: 26.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1}{\mathsf{fma}\left(v, v, -1\right)} - \frac{\left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1}{\mathsf{fma}\left(v, v, -1\right)} - \frac{\left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right)
double f(double v) {
        double r169295 = 1.0;
        double r169296 = 5.0;
        double r169297 = v;
        double r169298 = r169297 * r169297;
        double r169299 = r169296 * r169298;
        double r169300 = r169295 - r169299;
        double r169301 = r169298 - r169295;
        double r169302 = r169300 / r169301;
        double r169303 = acos(r169302);
        return r169303;
}
↓
double f(double v) {
        double r169304 = 1.0;
        double r169305 = v;
        double r169306 = -r169304;
        double r169307 = fma(r169305, r169305, r169306);
        double r169308 = r169304 / r169307;
        double r169309 = r169305 * r169305;
        double r169310 = 5.0;
        double r169311 = r169309 * r169310;
        double r169312 = r169311 / r169307;
        double r169313 = r169308 - r169312;
        double r169314 = acos(r169313);
        return r169314;
}

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. Simplified0.6

    \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - \left(v \cdot 5\right) \cdot v}{v \cdot v - 1}\right)}\]
  3. Using strategy rm

  4. Applied div-sub0.6

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

  5. Simplified0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))