Average Error: 0.0 → 0.0
Time: 3.1s
Precision: 64
\[0.7071100000000000163069557856942992657423 \cdot \left(\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \left(0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\right)\right)\]
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, \left(0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\right)\right)
double f(double x) {
        double r140009 = 0.70711;
        double r140010 = 2.30753;
        double r140011 = x;
        double r140012 = 0.27061;
        double r140013 = r140011 * r140012;
        double r140014 = r140010 + r140013;
        double r140015 = 1.0;
        double r140016 = 0.99229;
        double r140017 = 0.04481;
        double r140018 = r140011 * r140017;
        double r140019 = r140016 + r140018;
        double r140020 = r140011 * r140019;
        double r140021 = r140015 + r140020;
        double r140022 = r140014 / r140021;
        double r140023 = r140022 - r140011;
        double r140024 = r140009 * r140023;
        return r140024;
}
↓
double f(double x) {
        double r140025 = x;
        double r140026 = -r140025;
        double r140027 = 0.70711;
        double r140028 = 0.27061;
        double r140029 = 2.30753;
        double r140030 = fma(r140028, r140025, r140029);
        double r140031 = r140027 * r140030;
        double r140032 = 1.0;
        double r140033 = 0.04481;
        double r140034 = 0.99229;
        double r140035 = fma(r140033, r140025, r140034);
        double r140036 = 1.0;
        double r140037 = fma(r140025, r140035, r140036);
        double r140038 = r140032 / r140037;
        double r140039 = log1p(r140038);
        double r140040 = expm1(r140039);
        double r140041 = r140031 * r140040;
        double r140042 = fma(r140026, r140027, r140041);
        return r140042;
}

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 div-inv0.0

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

  6. Applied expm1-log1p-u0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \left(0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\right)}\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.7071100000000000163069557856942992657423, \left(0.7071100000000000163069557856942992657423 \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019352 +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)))