Average Error: 0.0 → 0.0
Time: 3.2s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}\right)\]
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)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}\right)
double f(double x) {
        double r109741 = 0.70711;
        double r109742 = 2.30753;
        double r109743 = x;
        double r109744 = 0.27061;
        double r109745 = r109743 * r109744;
        double r109746 = r109742 + r109745;
        double r109747 = 1.0;
        double r109748 = 0.99229;
        double r109749 = 0.04481;
        double r109750 = r109743 * r109749;
        double r109751 = r109748 + r109750;
        double r109752 = r109743 * r109751;
        double r109753 = r109747 + r109752;
        double r109754 = r109746 / r109753;
        double r109755 = r109754 - r109743;
        double r109756 = r109741 * r109755;
        return r109756;
}
↓
double f(double x) {
        double r109757 = x;
        double r109758 = -r109757;
        double r109759 = 0.70711;
        double r109760 = 0.27061;
        double r109761 = 2.30753;
        double r109762 = fma(r109760, r109757, r109761);
        double r109763 = r109759 * r109762;
        double r109764 = 0.04481;
        double r109765 = 0.99229;
        double r109766 = fma(r109764, r109757, r109765);
        double r109767 = 1.0;
        double r109768 = fma(r109757, r109766, r109767);
        double r109769 = 3.0;
        double r109770 = pow(r109768, r109769);
        double r109771 = cbrt(r109770);
        double r109772 = r109763 / r109771;
        double r109773 = fma(r109758, r109759, r109772);
        return r109773;
}

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 add-cbrt-cube0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}}\right)\]

Reproduce

herbie shell --seed 2020100 +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)))