Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)
double f(double x) {
        double r89861 = 0.70711;
        double r89862 = 2.30753;
        double r89863 = x;
        double r89864 = 0.27061;
        double r89865 = r89863 * r89864;
        double r89866 = r89862 + r89865;
        double r89867 = 1.0;
        double r89868 = 0.99229;
        double r89869 = 0.04481;
        double r89870 = r89863 * r89869;
        double r89871 = r89868 + r89870;
        double r89872 = r89863 * r89871;
        double r89873 = r89867 + r89872;
        double r89874 = r89866 / r89873;
        double r89875 = r89874 - r89863;
        double r89876 = r89861 * r89875;
        return r89876;
}
↓
double f(double x) {
        double r89877 = x;
        double r89878 = -r89877;
        double r89879 = 0.70711;
        double r89880 = 0.04481;
        double r89881 = 0.99229;
        double r89882 = fma(r89880, r89877, r89881);
        double r89883 = 1.0;
        double r89884 = fma(r89877, r89882, r89883);
        double r89885 = 0.27061;
        double r89886 = 2.30753;
        double r89887 = fma(r89885, r89877, r89886);
        double r89888 = r89884 / r89887;
        double r89889 = r89879 / r89888;
        double r89890 = fma(r89878, r89879, r89889);
        return r89890;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}\]
  3. Using strategy rm

  4. Applied associate-/l*0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \color{blue}{\frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))