Average Error: 0.0 → 0.0
Time: 2.2s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \mathsf{fma}\left(0.1913510371000000098717919172486290335655, x, 1.631677538299999952187135932035744190216\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]
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, \mathsf{fma}\left(0.1913510371000000098717919172486290335655, x, 1.631677538299999952187135932035744190216\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)
double f(double x) {
        double r81167 = 0.70711;
        double r81168 = 2.30753;
        double r81169 = x;
        double r81170 = 0.27061;
        double r81171 = r81169 * r81170;
        double r81172 = r81168 + r81171;
        double r81173 = 1.0;
        double r81174 = 0.99229;
        double r81175 = 0.04481;
        double r81176 = r81169 * r81175;
        double r81177 = r81174 + r81176;
        double r81178 = r81169 * r81177;
        double r81179 = r81173 + r81178;
        double r81180 = r81172 / r81179;
        double r81181 = r81180 - r81169;
        double r81182 = r81167 * r81181;
        return r81182;
}
↓
double f(double x) {
        double r81183 = x;
        double r81184 = -r81183;
        double r81185 = 0.70711;
        double r81186 = 0.1913510371;
        double r81187 = 1.6316775383;
        double r81188 = fma(r81186, r81183, r81187);
        double r81189 = 1.0;
        double r81190 = 0.04481;
        double r81191 = 0.99229;
        double r81192 = fma(r81190, r81183, r81191);
        double r81193 = 1.0;
        double r81194 = fma(r81183, r81192, r81193);
        double r81195 = r81189 / r81194;
        double r81196 = r81188 * r81195;
        double r81197 = fma(r81184, r81185, r81196);
        return r81197;
}

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. Taylor expanded around 0 0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{\color{blue}{0.1913510371000000098717919172486290335655 \cdot x + 1.631677538299999952187135932035744190216}}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{\color{blue}{\mathsf{fma}\left(0.1913510371000000098717919172486290335655, x, 1.631677538299999952187135932035744190216\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]
  5. Using strategy rm

  6. Applied div-inv0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \color{blue}{\mathsf{fma}\left(0.1913510371000000098717919172486290335655, x, 1.631677538299999952187135932035744190216\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \mathsf{fma}\left(0.1913510371000000098717919172486290335655, x, 1.631677538299999952187135932035744190216\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\]

Reproduce

herbie shell --seed 2019347 +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)))