Average Error: 0.0 → 0.0
Time: 2.4s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]
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 \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)
double f(double x) {
        double r93874 = 0.70711;
        double r93875 = 2.30753;
        double r93876 = x;
        double r93877 = 0.27061;
        double r93878 = r93876 * r93877;
        double r93879 = r93875 + r93878;
        double r93880 = 1.0;
        double r93881 = 0.99229;
        double r93882 = 0.04481;
        double r93883 = r93876 * r93882;
        double r93884 = r93881 + r93883;
        double r93885 = r93876 * r93884;
        double r93886 = r93880 + r93885;
        double r93887 = r93879 / r93886;
        double r93888 = r93887 - r93876;
        double r93889 = r93874 * r93888;
        return r93889;
}
↓
double f(double x) {
        double r93890 = x;
        double r93891 = -r93890;
        double r93892 = 0.70711;
        double r93893 = 0.27061;
        double r93894 = 2.30753;
        double r93895 = fma(r93893, r93890, r93894);
        double r93896 = r93892 * r93895;
        double r93897 = 0.04481;
        double r93898 = 0.99229;
        double r93899 = fma(r93897, r93890, r93898);
        double r93900 = 1.0;
        double r93901 = fma(r93890, r93899, r93900);
        double r93902 = r93896 / r93901;
        double r93903 = fma(r93891, r93892, r93902);
        return r93903;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020089 +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)))