Average Error: 0.0 → 0.0
Time: 29.8s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\frac{0.7071100000000000163069557856942992657423}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}} + \left(-x\right) \cdot 0.7071100000000000163069557856942992657423\]
0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)
↓
\frac{0.7071100000000000163069557856942992657423}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}} + \left(-x\right) \cdot 0.7071100000000000163069557856942992657423
double f(double x) {
        double r85718 = 0.70711;
        double r85719 = 2.30753;
        double r85720 = x;
        double r85721 = 0.27061;
        double r85722 = r85720 * r85721;
        double r85723 = r85719 + r85722;
        double r85724 = 1.0;
        double r85725 = 0.99229;
        double r85726 = 0.04481;
        double r85727 = r85720 * r85726;
        double r85728 = r85725 + r85727;
        double r85729 = r85720 * r85728;
        double r85730 = r85724 + r85729;
        double r85731 = r85723 / r85730;
        double r85732 = r85731 - r85720;
        double r85733 = r85718 * r85732;
        return r85733;
}
↓
double f(double x) {
        double r85734 = 0.70711;
        double r85735 = x;
        double r85736 = 0.04481;
        double r85737 = 0.99229;
        double r85738 = fma(r85736, r85735, r85737);
        double r85739 = 1.0;
        double r85740 = fma(r85735, r85738, r85739);
        double r85741 = 0.27061;
        double r85742 = 2.30753;
        double r85743 = fma(r85741, r85735, r85742);
        double r85744 = r85740 / r85743;
        double r85745 = r85734 / r85744;
        double r85746 = -r85735;
        double r85747 = r85746 * r85734;
        double r85748 = r85745 + r85747;
        return r85748;
}

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. Using strategy rm

  3. Applied sub-neg0.0

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

  4. Applied distribute-lft-in0.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +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)))