Average Error: 0.5 → 0.5
Time: 1.1m
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 r3708698 = 1.0;
        double r3708699 = 5.0;
        double r3708700 = v;
        double r3708701 = r3708700 * r3708700;
        double r3708702 = r3708699 * r3708701;
        double r3708703 = r3708698 - r3708702;
        double r3708704 = r3708701 - r3708698;
        double r3708705 = r3708703 / r3708704;
        double r3708706 = acos(r3708705);
        return r3708706;
}
↓
double f(double v) {
        double r3708707 = -5.0;
        double r3708708 = v;
        double r3708709 = r3708707 * r3708708;
        double r3708710 = 1.0;
        double r3708711 = fma(r3708709, r3708708, r3708710);
        double r3708712 = -1.0;
        double r3708713 = fma(r3708708, r3708708, r3708712);
        double r3708714 = r3708711 / r3708713;
        double r3708715 = acos(r3708714);
        return r3708715;
}

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. Simplified0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))