Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, 0.707110000000000016 \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]
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, 0.707110000000000016 \cdot \frac{\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 r118764 = 0.70711;
        double r118765 = 2.30753;
        double r118766 = x;
        double r118767 = 0.27061;
        double r118768 = r118766 * r118767;
        double r118769 = r118765 + r118768;
        double r118770 = 1.0;
        double r118771 = 0.99229;
        double r118772 = 0.04481;
        double r118773 = r118766 * r118772;
        double r118774 = r118771 + r118773;
        double r118775 = r118766 * r118774;
        double r118776 = r118770 + r118775;
        double r118777 = r118769 / r118776;
        double r118778 = r118777 - r118766;
        double r118779 = r118764 * r118778;
        return r118779;
}
↓
double f(double x) {
        double r118780 = x;
        double r118781 = -r118780;
        double r118782 = 0.70711;
        double r118783 = 0.27061;
        double r118784 = 2.30753;
        double r118785 = fma(r118783, r118780, r118784);
        double r118786 = 0.04481;
        double r118787 = 0.99229;
        double r118788 = fma(r118786, r118780, r118787);
        double r118789 = 1.0;
        double r118790 = fma(r118780, r118788, r118789);
        double r118791 = r118785 / r118790;
        double r118792 = r118782 * r118791;
        double r118793 = fma(r118781, r118782, r118792);
        return r118793;
}

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 *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 +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)))