Average Error: 0.0 → 0.0
Time: 2.4s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, 0.7071100000000000163069557856942992657423 \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]
0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, 0.7071100000000000163069557856942992657423 \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)
double f(double x) {
        double r96753 = 0.70711;
        double r96754 = 2.30753;
        double r96755 = x;
        double r96756 = 0.27061;
        double r96757 = r96755 * r96756;
        double r96758 = r96754 + r96757;
        double r96759 = 1.0;
        double r96760 = 0.99229;
        double r96761 = 0.04481;
        double r96762 = r96755 * r96761;
        double r96763 = r96760 + r96762;
        double r96764 = r96755 * r96763;
        double r96765 = r96759 + r96764;
        double r96766 = r96758 / r96765;
        double r96767 = r96766 - r96755;
        double r96768 = r96753 * r96767;
        return r96768;
}
↓
double f(double x) {
        double r96769 = x;
        double r96770 = -r96769;
        double r96771 = 0.70711;
        double r96772 = 0.27061;
        double r96773 = 2.30753;
        double r96774 = fma(r96772, r96769, r96773);
        double r96775 = 0.04481;
        double r96776 = 0.99229;
        double r96777 = fma(r96775, r96769, r96776);
        double r96778 = 1.0;
        double r96779 = fma(r96769, r96777, r96778);
        double r96780 = r96774 / r96779;
        double r96781 = r96771 * r96780;
        double r96782 = fma(r96770, r96771, r96781);
        return r96782;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}\right)\]

  5. Applied times-frac0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \color{blue}{\frac{0.7071100000000000163069557856942992657423}{1} \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}\right)\]

  6. Simplified0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \color{blue}{0.7071100000000000163069557856942992657423} \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, 0.7071100000000000163069557856942992657423 \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]

Reproduce

herbie shell --seed 2019346 +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)))