Average Error: 0.0 → 0.0
Time: 31.4s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\frac{0.7071100000000000163069557856942992657423}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}} + \left(-x\right) \cdot 0.7071100000000000163069557856942992657423\]
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 r94056 = 0.70711;
        double r94057 = 2.30753;
        double r94058 = x;
        double r94059 = 0.27061;
        double r94060 = r94058 * r94059;
        double r94061 = r94057 + r94060;
        double r94062 = 1.0;
        double r94063 = 0.99229;
        double r94064 = 0.04481;
        double r94065 = r94058 * r94064;
        double r94066 = r94063 + r94065;
        double r94067 = r94058 * r94066;
        double r94068 = r94062 + r94067;
        double r94069 = r94061 / r94068;
        double r94070 = r94069 - r94058;
        double r94071 = r94056 * r94070;
        return r94071;
}
↓
double f(double x) {
        double r94072 = 0.70711;
        double r94073 = x;
        double r94074 = 0.04481;
        double r94075 = 0.99229;
        double r94076 = fma(r94074, r94073, r94075);
        double r94077 = 1.0;
        double r94078 = fma(r94073, r94076, r94077);
        double r94079 = 0.27061;
        double r94080 = 2.30753;
        double r94081 = fma(r94079, r94073, r94080);
        double r94082 = r94078 / r94081;
        double r94083 = r94072 / r94082;
        double r94084 = -r94073;
        double r94085 = r94084 * r94072;
        double r94086 = r94083 + r94085;
        return r94086;
}

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)))