Jmat.Real.erf

Percentage Accurate: 79.0% → 99.4%

Time: 23.4s

Alternatives: 11

Speedup: 142.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + t_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + t_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x_m \cdot 1.128386358070218\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\left(\left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{3}}\right) - \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{2}}\right) - \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{4}}\right) - 0.254829592}{e^{{x_m}^{2}}}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-11)
   (/
    (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
    (+
     1e-18
     (-
      (pow (* x_m 1.128386358070218) 2.0)
      (* 1e-9 (* x_m 1.128386358070218)))))
   (+
    1.0
    (/
     (/
      (-
       (-
        (-
         (+
          (/ 0.284496736 (fma 0.3275911 x_m 1.0))
          (/ 1.453152027 (pow (fma 0.3275911 x_m 1.0) 3.0)))
         (/ 1.421413741 (pow (fma 0.3275911 x_m 1.0) 2.0)))
        (/ 1.061405429 (pow (fma 0.3275911 x_m 1.0) 4.0)))
       0.254829592)
      (exp (pow x_m 2.0)))
     (fma 0.3275911 x_m 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - (1e-9 * (x_m * 1.128386358070218))));
	} else {
		tmp = 1.0 + (((((((0.284496736 / fma(0.3275911, x_m, 1.0)) + (1.453152027 / pow(fma(0.3275911, x_m, 1.0), 3.0))) - (1.421413741 / pow(fma(0.3275911, x_m, 1.0), 2.0))) - (1.061405429 / pow(fma(0.3275911, x_m, 1.0), 4.0))) - 0.254829592) / exp(pow(x_m, 2.0))) / fma(0.3275911, x_m, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(1e-9 * Float64(x_m * 1.128386358070218)))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.284496736 / fma(0.3275911, x_m, 1.0)) + Float64(1.453152027 / (fma(0.3275911, x_m, 1.0) ^ 3.0))) - Float64(1.421413741 / (fma(0.3275911, x_m, 1.0) ^ 2.0))) - Float64(1.061405429 / (fma(0.3275911, x_m, 1.0) ^ 4.0))) - 0.254829592) / exp((x_m ^ 2.0))) / fma(0.3275911, x_m, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(1e-9 * N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(N[(N[(0.284496736 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.421413741 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.061405429 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.99999999999999939e-12

   1. Initial program 57.5%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.1%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 57.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   9. Step-by-step derivation

      1. flip3-+ 97.1%

         \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(1.128386358070218 \cdot x\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)}} \]

      2. metadata-eval 97.1%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(1.128386358070218 \cdot x\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      3. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(x \cdot 1.128386358070218\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      4. unpow-prod-down 97.1%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      5. metadata-eval 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      6. metadata-eval 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      7. pow2 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(1.128386358070218 \cdot x\right)}^{2}} - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      8. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\color{blue}{\left(x \cdot 1.128386358070218\right)}}^{2} - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      9. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \color{blue}{\left(x \cdot 1.128386358070218\right)}\right)} \]

   10. Applied egg-rr 97.1%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 97.1%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   12. Simplified 97.1%

       \[\leadsto \color{blue}{\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 9.99999999999999939e-12 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-11}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\left(\left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) - \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) - \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) - 0.254829592}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x_m \cdot 1.128386358070218\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{e^{{x_m}^{2}}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-11)
   (/
    (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
    (+
     1e-18
     (-
      (pow (* x_m 1.128386358070218) 2.0)
      (* 1e-9 (* x_m 1.128386358070218)))))
   (exp
    (log1p
     (/
      (/
       (-
        -0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0))
      (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - (1e-9 * (x_m * 1.128386358070218))));
	} else {
		tmp = exp(log1p((((-0.254829592 - ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) / exp(pow(x_m, 2.0)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(1e-9 * Float64(x_m * 1.128386358070218)))));
	else
		tmp = exp(log1p(Float64(Float64(Float64(-0.254829592 - Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) / exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(1e-9 * N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[1 + N[(N[(N[(-0.254829592 - N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.99999999999999939e-12

   1. Initial program 57.5%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.1%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 57.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   9. Step-by-step derivation

      1. flip3-+ 97.1%

         \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(1.128386358070218 \cdot x\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)}} \]

      2. metadata-eval 97.1%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(1.128386358070218 \cdot x\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      3. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(x \cdot 1.128386358070218\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      4. unpow-prod-down 97.1%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      5. metadata-eval 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      6. metadata-eval 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      7. pow2 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(1.128386358070218 \cdot x\right)}^{2}} - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      8. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\color{blue}{\left(x \cdot 1.128386358070218\right)}}^{2} - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      9. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \color{blue}{\left(x \cdot 1.128386358070218\right)}\right)} \]

   10. Applied egg-rr 97.1%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 97.1%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   12. Simplified 97.1%

       \[\leadsto \color{blue}{\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 9.99999999999999939e-12 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. log1p-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. add-exp-log 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. Applied egg-rr 99.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   7. Step-by-step derivation

      1. fma-udef 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. associate--l+ 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. metadata-eval 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. Simplified 99.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. log1p-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. add-exp-log 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. fma-udef 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   12. Simplified 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   14. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot e^{-{x}^{2}}\right)}} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-11}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x_m \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := 1 + \left|x_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x_m\right| \leq 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x_m \cdot 1.128386358070218\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x_m \cdot \left(-x_m\right)} \cdot \left(\frac{1}{t_2} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right) \cdot \frac{-1}{t_2} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ 1.0 (* (fabs x_m) 0.3275911))))
   (if (<= (fabs x_m) 1e-11)
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       1e-18
       (-
        (pow (* x_m 1.128386358070218) 2.0)
        (* 1e-9 (* x_m 1.128386358070218)))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 t_2)
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0))))))
          (/ -1.0 t_2))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (fabs(x_m) * 0.3275911);
	double tmp;
	if (fabs(x_m) <= 1e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - (1e-9 * (x_m * 1.128386358070218))));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / t_2) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))) * (-1.0 / t_2)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (x_m * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    t_2 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    if (abs(x_m) <= 1d-11) then
        tmp = (((x_m ** 3.0d0) * 1.436724444676459d0) + 1d-27) / (1d-18 + (((x_m * 1.128386358070218d0) ** 2.0d0) - (1d-9 * (x_m * 1.128386358070218d0))))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((1.0d0 / t_2) * ((((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0)))))) * ((-1.0d0) / t_2)) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double tmp;
	if (Math.abs(x_m) <= 1e-11) {
		tmp = ((Math.pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + (Math.pow((x_m * 1.128386358070218), 2.0) - (1e-9 * (x_m * 1.128386358070218))));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((1.0 / t_2) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))) * (-1.0 / t_2)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (x_m * 0.3275911)
	t_1 = 1.0 / t_0
	t_2 = 1.0 + (math.fabs(x_m) * 0.3275911)
	tmp = 0
	if math.fabs(x_m) <= 1e-11:
		tmp = ((math.pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + (math.pow((x_m * 1.128386358070218), 2.0) - (1e-9 * (x_m * 1.128386358070218))))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((1.0 / t_2) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))) * (-1.0 / t_2)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(x_m * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	tmp = 0.0
	if (abs(x_m) <= 1e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(1e-9 * Float64(x_m * 1.128386358070218)))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / t_2) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))) * Float64(-1.0 / t_2)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (x_m * 0.3275911);
	t_1 = 1.0 / t_0;
	t_2 = 1.0 + (abs(x_m) * 0.3275911);
	tmp = 0.0;
	if (abs(x_m) <= 1e-11)
		tmp = (((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + (((x_m * 1.128386358070218) ^ 2.0) - (1e-9 * (x_m * 1.128386358070218))));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / t_2) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))) * (-1.0 / t_2)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(1e-9 * N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.99999999999999939e-12

   1. Initial program 57.5%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.1%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 57.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   9. Step-by-step derivation

      1. flip3-+ 97.1%

         \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(1.128386358070218 \cdot x\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)}} \]

      2. metadata-eval 97.1%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(1.128386358070218 \cdot x\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      3. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(x \cdot 1.128386358070218\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      4. unpow-prod-down 97.1%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      5. metadata-eval 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      6. metadata-eval 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right) - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      7. pow2 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(1.128386358070218 \cdot x\right)}^{2}} - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      8. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\color{blue}{\left(x \cdot 1.128386358070218\right)}}^{2} - 10^{-9} \cdot \left(1.128386358070218 \cdot x\right)\right)} \]

      9. *-commutative 97.1%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \color{blue}{\left(x \cdot 1.128386358070218\right)}\right)} \]

   10. Applied egg-rr 97.1%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 97.1%

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   12. Simplified 97.1%

       \[\leadsto \color{blue}{\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 9.99999999999999939e-12 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. log1p-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. add-exp-log 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. Applied egg-rr 99.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   7. Step-by-step derivation

      1. fma-udef 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. associate--l+ 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. metadata-eval 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. Simplified 99.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. log1p-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. add-exp-log 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-udef 99.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 50.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 99.1%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. fma-udef 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   12. Simplified 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. expm1-udef 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. log1p-udef 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. add-exp-log 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. +-commutative 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. fma-udef 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 50.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 50.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   14. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   15. Step-by-step derivation

       1. fma-udef 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 99.1%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   16. Simplified 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-11}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + x_m \cdot 0.3275911}\\ \mathbf{if}\;x_m \leq 0.75:\\ \;\;\;\;10^{-9} + \left({x_m}^{3} \cdot -0.37545125292247583 + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot 1.029667143\right)\right) \cdot \frac{-1}{1 + \left|x_m\right| \cdot 0.3275911}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x_m 0.3275911)))))
   (if (<= x_m 0.75)
     (+
      1e-9
      (+
       (* (pow x_m 3.0) -0.37545125292247583)
       (+
        (* (pow x_m 2.0) -0.00011824294398844343)
        (* x_m 1.128386358070218))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+ 0.254829592 (* t_0 (+ -0.284496736 (* t_0 1.029667143))))
        (/ -1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	double tmp;
	if (x_m <= 0.75) {
		tmp = 1e-9 + ((pow(x_m, 3.0) * -0.37545125292247583) + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * 1.029667143)))) * (-1.0 / (1.0 + (fabs(x_m) * 0.3275911)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (x_m * 0.3275911d0))
    if (x_m <= 0.75d0) then
        tmp = 1d-9 + (((x_m ** 3.0d0) * (-0.37545125292247583d0)) + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * 1.029667143d0)))) * ((-1.0d0) / (1.0d0 + (abs(x_m) * 0.3275911d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	double tmp;
	if (x_m <= 0.75) {
		tmp = 1e-9 + ((Math.pow(x_m, 3.0) * -0.37545125292247583) + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * 1.029667143)))) * (-1.0 / (1.0 + (Math.abs(x_m) * 0.3275911)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911))
	tmp = 0
	if x_m <= 0.75:
		tmp = 1e-9 + ((math.pow(x_m, 3.0) * -0.37545125292247583) + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * 1.029667143)))) * (-1.0 / (1.0 + (math.fabs(x_m) * 0.3275911)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))
	tmp = 0.0
	if (x_m <= 0.75)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 3.0) * -0.37545125292247583) + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * 1.029667143)))) * Float64(-1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	tmp = 0.0;
	if (x_m <= 0.75)
		tmp = 1e-9 + (((x_m ^ 3.0) * -0.37545125292247583) + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * 1.029667143)))) * (-1.0 / (1.0 + (abs(x_m) * 0.3275911)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.75], N[(1e-9 + N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * 1.029667143), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 71.3%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 69.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 66.6%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 0.75 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \color{blue}{\frac{1 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{\color{blue}{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. +-commutative 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fma-udef 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-udef 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-cube-cbrt 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right) \cdot \sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. pow3 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{{\left(\sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)}^{3}}\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. Applied egg-rr 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{{\left(\sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}}\right)\right)\right) \cdot e^{-x \cdot x} \]

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{1.029667143}\right)\right)\right) \cdot e^{-x \cdot x} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. log1p-udef 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. add-exp-log 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. +-commutative 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fma-udef 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   9. Applied egg-rr 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]
   10. Step-by-step derivation

       1. fma-udef 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   11. Simplified 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. expm1-udef 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. log1p-udef 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. add-exp-log 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. +-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. fma-udef 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   13. Applied egg-rr 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]
   14. Step-by-step derivation

       1. fma-udef 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]

   15. Simplified 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot 1.029667143\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot 1.029667143\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;10^{-9} + \left({x_m}^{3} \cdot -0.37545125292247583 + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 + \left(-1 - \frac{\frac{0.7778892405807117}{x_m}}{e^{{x_m}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (+
    1e-9
    (+
     (* (pow x_m 3.0) -0.37545125292247583)
     (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218))))
   (+
    1.0
    (+ 1.0 (- -1.0 (/ (/ 0.7778892405807117 x_m) (exp (pow x_m 2.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1e-9 + ((pow(x_m, 3.0) * -0.37545125292247583) + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / exp(pow(x_m, 2.0)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = 1d-9 + (((x_m ** 3.0d0) * (-0.37545125292247583d0)) + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (1.0d0 + ((-1.0d0) - ((0.7778892405807117d0 / x_m) / exp((x_m ** 2.0d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1e-9 + ((Math.pow(x_m, 3.0) * -0.37545125292247583) + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / Math.exp(Math.pow(x_m, 2.0)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 1e-9 + ((math.pow(x_m, 3.0) * -0.37545125292247583) + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / math.exp(math.pow(x_m, 2.0)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 3.0) * -0.37545125292247583) + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(Float64(0.7778892405807117 / x_m) / exp((x_m ^ 2.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 1e-9 + (((x_m ^ 3.0) * -0.37545125292247583) + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / exp((x_m ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(1e-9 + N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 + N[(-1.0 - N[(N[(0.7778892405807117 / x$95$m), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.3%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 69.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 66.6%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 - 0.7778892405807117 \cdot \frac{1}{x \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto 1 - \color{blue}{\frac{0.7778892405807117 \cdot 1}{x \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 99.9%

         \[\leadsto 1 - \frac{\color{blue}{0.7778892405807117}}{x \cdot e^{{x}^{2}}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 99.9%

          \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)\right)} \]

       2. expm1-udef 99.9%

          \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)} - 1\right)} \]

       3. log1p-udef 99.9%

          \[\leadsto 1 - \left(e^{\color{blue}{\log \left(1 + \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)}} - 1\right) \]

       4. add-exp-log 99.9%

          \[\leadsto 1 - \left(\color{blue}{\left(1 + \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)} - 1\right) \]

       5. associate-/r* 99.9%

          \[\leadsto 1 - \left(\left(1 + \color{blue}{\frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}}\right) - 1\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto 1 - \color{blue}{\left(\left(1 + \frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 + \left(-1 - \frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.86:\\ \;\;\;\;10^{-9} + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 + \left(-1 - \frac{\frac{0.7778892405807117}{x_m}}{e^{{x_m}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.86)
   (+
    1e-9
    (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218)))
   (+
    1.0
    (+ 1.0 (- -1.0 (/ (/ 0.7778892405807117 x_m) (exp (pow x_m 2.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.86) {
		tmp = 1e-9 + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / exp(pow(x_m, 2.0)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.86d0) then
        tmp = 1d-9 + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 + (1.0d0 + ((-1.0d0) - ((0.7778892405807117d0 / x_m) / exp((x_m ** 2.0d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.86) {
		tmp = 1e-9 + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / Math.exp(Math.pow(x_m, 2.0)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.86:
		tmp = 1e-9 + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / math.exp(math.pow(x_m, 2.0)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.86)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(Float64(0.7778892405807117 / x_m) / exp((x_m ^ 2.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.86)
		tmp = 1e-9 + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	else
		tmp = 1.0 + (1.0 + (-1.0 - ((0.7778892405807117 / x_m) / exp((x_m ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.86], N[(1e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 + N[(-1.0 - N[(N[(0.7778892405807117 / x$95$m), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 71.3%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 69.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]

   if 0.859999999999999987 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 - 0.7778892405807117 \cdot \frac{1}{x \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto 1 - \color{blue}{\frac{0.7778892405807117 \cdot 1}{x \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 99.9%

         \[\leadsto 1 - \frac{\color{blue}{0.7778892405807117}}{x \cdot e^{{x}^{2}}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 99.9%

          \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)\right)} \]

       2. expm1-udef 99.9%

          \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)} - 1\right)} \]

       3. log1p-udef 99.9%

          \[\leadsto 1 - \left(e^{\color{blue}{\log \left(1 + \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)}} - 1\right) \]

       4. add-exp-log 99.9%

          \[\leadsto 1 - \left(\color{blue}{\left(1 + \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\right)} - 1\right) \]

       5. associate-/r* 99.9%

          \[\leadsto 1 - \left(\left(1 + \color{blue}{\frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}}\right) - 1\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto 1 - \color{blue}{\left(\left(1 + \frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;10^{-9} + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 + \left(-1 - \frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.86:\\ \;\;\;\;10^{-9} + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.86)
   (+
    1e-9
    (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218)))
   (- 1.0 (/ 0.7778892405807117 (* x_m (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.86) {
		tmp = 1e-9 + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.86d0) then
        tmp = 1d-9 + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x_m * exp((x_m ** 2.0d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.86) {
		tmp = 1e-9 + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * Math.exp(Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.86:
		tmp = 1e-9 + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x_m * math.exp(math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.86)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m * exp((x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.86)
		tmp = 1e-9 + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	else
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp((x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.86], N[(1e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 71.3%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 69.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]

   if 0.859999999999999987 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 - 0.7778892405807117 \cdot \frac{1}{x \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto 1 - \color{blue}{\frac{0.7778892405807117 \cdot 1}{x \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 99.9%

         \[\leadsto 1 - \frac{\color{blue}{0.7778892405807117}}{x \cdot e^{{x}^{2}}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;10^{-9} + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.9:\\ \;\;\;\;10^{-9} + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9)
   (+
    1e-9
    (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = 1d-9 + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = 1e-9 + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = 1e-9 + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(1e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 71.3%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 69.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.9:\\ \;\;\;\;10^{-9} + x_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9) (+ 1e-9 (* x_m 1.128386358070218)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 71.3%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 69.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 142.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) 1e-9 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.8e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.2%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 68.9%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around 0 68.6%

      \[\leadsto \color{blue}{10^{-9}} \]

   if 2.79999999999999996e-5 < x

   1. Initial program 99.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   8. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.5% accurate, 856.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 10^{-9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1e-9)

C

x_m = fabs(x);
double code(double x_m) {
	return 1e-9;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1d-9
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return 1e-9
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1e-9;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1e-9

Derivation

1. Initial program 78.4%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

3. Simplified 78.4%

   \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 76.6%

   \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

7. Simplified 77.9%

   \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right)\right)\right)}{e^{{x}^{2}}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}} \]

8. Taylor expanded in x around 0 54.2%

   \[\leadsto \color{blue}{10^{-9}} \]

9. Final simplification 54.2%

   \[\leadsto 10^{-9} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))