Average Error: 0.0 → 0.0
Time: 2.9s
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 r117161 = 0.70711;
        double r117162 = 2.30753;
        double r117163 = x;
        double r117164 = 0.27061;
        double r117165 = r117163 * r117164;
        double r117166 = r117162 + r117165;
        double r117167 = 1.0;
        double r117168 = 0.99229;
        double r117169 = 0.04481;
        double r117170 = r117163 * r117169;
        double r117171 = r117168 + r117170;
        double r117172 = r117163 * r117171;
        double r117173 = r117167 + r117172;
        double r117174 = r117166 / r117173;
        double r117175 = r117174 - r117163;
        double r117176 = r117161 * r117175;
        return r117176;
}
↓
double f(double x) {
        double r117177 = x;
        double r117178 = -r117177;
        double r117179 = 0.70711;
        double r117180 = 0.04481;
        double r117181 = 0.99229;
        double r117182 = fma(r117180, r117177, r117181);
        double r117183 = 1.0;
        double r117184 = fma(r117177, r117182, r117183);
        double r117185 = 0.27061;
        double r117186 = 2.30753;
        double r117187 = fma(r117185, r117177, r117186);
        double r117188 = r117184 / r117187;
        double r117189 = r117179 / r117188;
        double r117190 = fma(r117178, r117179, r117189);
        return r117190;
}

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 2020060 +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)))