Average Error: 0.0 → 0.1
Time: 2.8s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \frac{\frac{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}\right)\]
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, \frac{\frac{0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}}\right)
double f(double x) {
        double r115435 = 0.70711;
        double r115436 = 2.30753;
        double r115437 = x;
        double r115438 = 0.27061;
        double r115439 = r115437 * r115438;
        double r115440 = r115436 + r115439;
        double r115441 = 1.0;
        double r115442 = 0.99229;
        double r115443 = 0.04481;
        double r115444 = r115437 * r115443;
        double r115445 = r115442 + r115444;
        double r115446 = r115437 * r115445;
        double r115447 = r115441 + r115446;
        double r115448 = r115440 / r115447;
        double r115449 = r115448 - r115437;
        double r115450 = r115435 * r115449;
        return r115450;
}
↓
double f(double x) {
        double r115451 = x;
        double r115452 = -r115451;
        double r115453 = 0.70711;
        double r115454 = 0.27061;
        double r115455 = 2.30753;
        double r115456 = fma(r115454, r115451, r115455);
        double r115457 = r115453 * r115456;
        double r115458 = 0.04481;
        double r115459 = 0.99229;
        double r115460 = fma(r115458, r115451, r115459);
        double r115461 = 1.0;
        double r115462 = fma(r115451, r115460, r115461);
        double r115463 = sqrt(r115462);
        double r115464 = r115457 / r115463;
        double r115465 = r115464 / r115463;
        double r115466 = fma(r115452, r115453, r115465);
        return r115466;
}

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 add-sqr-sqrt0.1

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

  5. Applied associate-/r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020002 +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)))