Average Error: 0.6 → 0.6
Time: 29.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)
double f(double v) {
        double r4775928 = 1.0;
        double r4775929 = 5.0;
        double r4775930 = v;
        double r4775931 = r4775930 * r4775930;
        double r4775932 = r4775929 * r4775931;
        double r4775933 = r4775928 - r4775932;
        double r4775934 = r4775931 - r4775928;
        double r4775935 = r4775933 / r4775934;
        double r4775936 = acos(r4775935);
        return r4775936;
}
↓
double f(double v) {
        double r4775937 = -5.0;
        double r4775938 = v;
        double r4775939 = r4775937 * r4775938;
        double r4775940 = 1.0;
        double r4775941 = fma(r4775939, r4775938, r4775940);
        double r4775942 = -1.0;
        double r4775943 = fma(r4775938, r4775938, r4775942);
        double r4775944 = r4775941 / r4775943;
        double r4775945 = acos(r4775944);
        return r4775945;
}

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

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

  4. Applied expm1-log1p-u0.6

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

  5. Taylor expanded around inf 0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))