Jmat.Real.erf

Percentage Accurate: 79.2% → 99.6%

Time: 45.6s

Alternatives: 9

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: 79.2% 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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-{x_m}^{2}}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)\\ t_2 := 0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{t_1 \cdot \left(2.111650813574209 + \mathsf{fma}\left(1.126581484710674, {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{-2}, \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right)\right)}}{t_1}}{t_1}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x_m \cdot 1.128386358070218\right) \cdot \left(x_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{t_0 \cdot t_2}{t_1}\right)}^{2}}{\mathsf{fma}\left(t_0, \frac{t_2}{t_1}, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (- (pow x_m 2.0))))
        (t_1 (fma 0.3275911 (fabs x_m) 1.0))
        (t_2
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (fma
                1.1957597040827899
                (pow (fma 0.3275911 x_m 1.0) -3.0)
                -3.0685496600615605)
               (*
                t_1
                (+
                 2.111650813574209
                 (fma
                  1.126581484710674
                  (pow (fma 0.3275911 x_m 1.0) -2.0)
                  (/ 1.5423834506201546 (fma 0.3275911 x_m 1.0)))))))
             t_1))
           t_1))))
   (if (<= (fabs x_m) 2e-6)
     (/
      (fma (pow x_m 3.0) 1.436724444676459 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
     (/ (- 1.0 (pow (/ (* t_0 t_2) t_1) 2.0)) (fma t_0 (/ t_2 t_1) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(-pow(x_m, 2.0));
	double t_1 = fma(0.3275911, fabs(x_m), 1.0);
	double t_2 = 0.254829592 + ((-0.284496736 + ((1.421413741 + (fma(1.1957597040827899, pow(fma(0.3275911, x_m, 1.0), -3.0), -3.0685496600615605) / (t_1 * (2.111650813574209 + fma(1.126581484710674, pow(fma(0.3275911, x_m, 1.0), -2.0), (1.5423834506201546 / fma(0.3275911, x_m, 1.0))))))) / t_1)) / t_1);
	double tmp;
	if (fabs(x_m) <= 2e-6) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = (1.0 - pow(((t_0 * t_2) / t_1), 2.0)) / fma(t_0, (t_2 / t_1), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = exp(Float64(-(x_m ^ 2.0)))
	t_1 = fma(0.3275911, abs(x_m), 1.0)
	t_2 = Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(fma(1.1957597040827899, (fma(0.3275911, x_m, 1.0) ^ -3.0), -3.0685496600615605) / Float64(t_1 * Float64(2.111650813574209 + fma(1.126581484710674, (fma(0.3275911, x_m, 1.0) ^ -2.0), Float64(1.5423834506201546 / fma(0.3275911, x_m, 1.0))))))) / t_1)) / t_1))
	tmp = 0.0
	if (abs(x_m) <= 2e-6)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(Float64(1.0 - (Float64(Float64(t_0 * t_2) / t_1) ^ 2.0)) / fma(t_0, Float64(t_2 / t_1), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(1.1957597040827899 * N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -3.0], $MachinePrecision] + -3.0685496600615605), $MachinePrecision] / N[(t$95$1 * N[(2.111650813574209 + N[(1.126581484710674 * N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -2.0], $MachinePrecision] + N[(1.5423834506201546 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-6], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[(t$95$0 * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$2 / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999991e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.0%

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

      1. distribute-rgt-neg-in 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in x around 0 97.5%

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

       1. *-commutative 97.5%

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

   11. Simplified 97.5%

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

       1. flip3-+ 97.5%

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

       2. metadata-eval 97.5%

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

       3. metadata-eval 97.5%

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

       4. pow2 97.5%

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

   13. Applied egg-rr 97.5%

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

       1. +-commutative 97.5%

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

       2. cube-prod 97.5%

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

       3. fma-def 97.5%

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

       4. metadata-eval 97.5%

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

       5. cancel-sign-sub-inv 97.5%

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

       6. unpow2 97.5%

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

       7. distribute-rgt-out 97.5%

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

       8. metadata-eval 97.5%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   15. Simplified 97.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.99999999999999991e-6 < (fabs.f64 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. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. fma-udef 99.7%

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

      3. flip3-+ 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\color{blue}{-3.0685496600615605} + {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. add-sqr-sqrt 55.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right)}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. fabs-sqr 55.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right)}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 99.7%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. metadata-eval 99.7%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}}{\color{blue}{2.111650813574209} + \left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. Applied egg-rr 99.7%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   7. Step-by-step derivation

      1. cube-div 99.7%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + \color{blue}{\frac{{1.061405429}^{3}}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. metadata-eval 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + \frac{\color{blue}{1.1957597040827899}}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. associate-*r/ 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - \color{blue}{\frac{-1.453152027 \cdot 1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. metadata-eval 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - \frac{\color{blue}{-1.5423834506201546}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. Simplified 99.8%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - \frac{-1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{1 - {\left(e^{-{x}^{2}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{\left(2.111650813574209 + 1.126581484710674 \cdot {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}\right) - \frac{-1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}}{1 + e^{-{x}^{2}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{\left(2.111650813574209 + 1.126581484710674 \cdot {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}\right) - \frac{-1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}} \]

   12. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{e^{-{x}^{2}} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(2.111650813574209 + \mathsf{fma}\left(1.126581484710674, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}}{\mathsf{fma}\left(e^{-{x}^{2}}, \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(2.111650813574209 + \mathsf{fma}\left(1.126581484710674, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{e^{-{x}^{2}} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(2.111650813574209 + \mathsf{fma}\left(1.126581484710674, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}}{\mathsf{fma}\left(e^{-{x}^{2}}, \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(1.1957597040827899, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -3.0685496600615605\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(2.111650813574209 + \mathsf{fma}\left(1.126581484710674, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x_m \cdot 1.128386358070218\right) \cdot \left(x_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x_m \cdot \left(-x_m\right)} \cdot \left(e^{-\mathsf{log1p}\left(t_0\right)} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x_m \cdot 0.3275911}\right)\right)\right) \cdot \frac{-1}{1 + \log \left(1 + \mathsf{expm1}\left(x_m \cdot 0.3275911\right)\right)} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (/ 1.0 (+ 1.0 t_0))))
   (if (<= (fabs x_m) 1e-10)
     (/
      (fma (pow x_m 3.0) 1.436724444676459 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (exp (- (log1p t_0)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              t_1
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x_m 0.3275911))))))))
          (/ -1.0 (+ 1.0 (log (+ 1.0 (expm1 (* x_m 0.3275911)))))))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if (fabs(x_m) <= 1e-10) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (exp(-log1p(t_0)) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))))))) * (-1.0 / (1.0 + log((1.0 + expm1((x_m * 0.3275911))))))) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(1.0 + t_0))
	tmp = 0.0
	if (abs(x_m) <= 1e-10)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(exp(Float64(-log1p(t_0))) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(x_m * 0.3275911)))))))) * Float64(-1.0 / Float64(1.0 + log(Float64(1.0 + expm1(Float64(x_m * 0.3275911))))))) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-10], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[(-N[Log[1 + t$95$0], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[Log[N[(1.0 + N[(Exp[N[(x$95$m * 0.3275911), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000004e-10

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 57.1%

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

      1. distribute-rgt-neg-in 57.1%

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

   8. Simplified 57.1%

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

   9. Taylor expanded in x around 0 97.8%

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

       1. *-commutative 97.8%

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

   11. Simplified 97.8%

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

       1. flip3-+ 97.8%

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

       2. metadata-eval 97.8%

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

       3. metadata-eval 97.8%

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

       4. pow2 97.8%

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

   13. Applied egg-rr 97.8%

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

       1. +-commutative 97.8%

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

       2. cube-prod 97.8%

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

       3. fma-def 97.8%

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

       4. metadata-eval 97.8%

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

       5. cancel-sign-sub-inv 97.8%

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

       6. unpow2 97.8%

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

       7. distribute-rgt-out 97.8%

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

       8. metadata-eval 97.8%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   15. Simplified 97.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.00000000000000004e-10 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.5%

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

      3. log1p-udef 99.5%

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

      4. add-exp-log 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-udef 99.5%

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

      7. add-sqr-sqrt 55.1%

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

      8. fabs-sqr 55.1%

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

      9. add-sqr-sqrt 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. fma-udef 99.5%

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

      2. associate--l+ 99.5%

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

      3. metadata-eval 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 99.5%

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

   8. Simplified 99.5%

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

      1. log1p-expm1-u 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. log1p-udef 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. add-sqr-sqrt 55.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fabs-sqr 55.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. add-sqr-sqrt 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \color{blue}{x}\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. add-exp-log 99.5%

          \[\leadsto 1 - \left(\color{blue}{e^{\log \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log-rec 99.5%

          \[\leadsto 1 - \left(e^{\color{blue}{-\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. log1p-udef 99.5%

          \[\leadsto 1 - \left(e^{-\color{blue}{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)}} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   12. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\color{blue}{e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)}} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(e^{-\mathsf{log1p}\left(\left|x\right| \cdot 0.3275911\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right) \cdot \frac{-1}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := 1 + x_m \cdot 0.3275911\\ \mathbf{if}\;\left|x_m\right| \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x_m \cdot 1.128386358070218\right) \cdot \left(x_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x_m \cdot \left(-x_m\right)} \cdot \left(\frac{1}{t_0} \cdot \left(\frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(x_m \cdot 0.3275911\right)\right)} \cdot \left(\left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{t_1}\right) \cdot \frac{1}{t_1}\right) \cdot \frac{-1}{t_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (+ 1.0 (* x_m 0.3275911))))
   (if (<= (fabs x_m) 1e-10)
     (/
      (fma (pow x_m 3.0) 1.436724444676459 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 t_0)
        (-
         (*
          (/ 1.0 (+ 1.0 (log (+ 1.0 (expm1 (* x_m 0.3275911))))))
          (-
           (*
            (+
             1.421413741
             (* (+ -1.453152027 (/ 1.061405429 t_1)) (/ 1.0 t_1)))
            (/ -1.0 t_0))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 + (x_m * 0.3275911);
	double tmp;
	if (fabs(x_m) <= 1e-10) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / t_0) * (((1.0 / (1.0 + log((1.0 + expm1((x_m * 0.3275911)))))) * (((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) * (1.0 / t_1))) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 + Float64(x_m * 0.3275911))
	tmp = 0.0
	if (abs(x_m) <= 1e-10)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / t_0) * Float64(Float64(Float64(1.0 / Float64(1.0 + log(Float64(1.0 + expm1(Float64(x_m * 0.3275911)))))) * Float64(Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) * Float64(1.0 / t_1))) * Float64(-1.0 / t_0)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-10], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $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$0), $MachinePrecision] * N[(N[(N[(1.0 / N[(1.0 + N[Log[N[(1.0 + N[(Exp[N[(x$95$m * 0.3275911), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000004e-10

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 57.1%

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

      1. distribute-rgt-neg-in 57.1%

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

   8. Simplified 57.1%

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

   9. Taylor expanded in x around 0 97.8%

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

       1. *-commutative 97.8%

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

   11. Simplified 97.8%

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

       1. flip3-+ 97.8%

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

       2. metadata-eval 97.8%

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

       3. metadata-eval 97.8%

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

       4. pow2 97.8%

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

   13. Applied egg-rr 97.8%

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

       1. +-commutative 97.8%

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

       2. cube-prod 97.8%

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

       3. fma-def 97.8%

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

       4. metadata-eval 97.8%

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

       5. cancel-sign-sub-inv 97.8%

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

       6. unpow2 97.8%

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

       7. distribute-rgt-out 97.8%

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

       8. metadata-eval 97.8%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   15. Simplified 97.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.00000000000000004e-10 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.5%

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

      3. log1p-udef 99.5%

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

      4. add-exp-log 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-udef 99.5%

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

      7. add-sqr-sqrt 55.1%

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

      8. fabs-sqr 55.1%

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

      9. add-sqr-sqrt 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. fma-udef 99.5%

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

      2. associate--l+ 99.5%

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

      3. metadata-eval 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 99.5%

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

   8. Simplified 99.5%

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

      1. log1p-expm1-u 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. log1p-udef 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. add-sqr-sqrt 55.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fabs-sqr 55.2%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. add-sqr-sqrt 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \color{blue}{x}\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 99.5%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. expm1-udef 99.5%

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

       3. log1p-udef 99.5%

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

       4. add-exp-log 99.5%

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

       5. +-commutative 99.5%

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

       6. fma-udef 99.5%

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

       7. add-sqr-sqrt 55.1%

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

       8. fabs-sqr 55.1%

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

       9. add-sqr-sqrt 99.5%

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

   12. Applied egg-rr 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   13. Step-by-step derivation

       1. fma-udef 99.5%

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

       2. associate--l+ 99.5%

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

       3. metadata-eval 99.5%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 99.5%

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

   14. Simplified 99.5%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)} \cdot \left(\left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right) \cdot \frac{1}{1 + x \cdot 0.3275911}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911)))
        (t_1 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x_m) 1e-10)
     (/
      (fma (pow x_m 3.0) 1.436724444676459 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+
         0.254829592
         (*
          t_2
          (+
           -0.284496736
           (*
            t_2
            (+
             1.421413741
             (* (+ -1.453152027 (/ 1.061405429 t_0)) (/ 1.0 t_0)))))))
        (/ -1.0 t_1)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double t_1 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (fabs(x_m) <= 1e-10) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) * (1.0 / t_0))))))) * (-1.0 / t_1)));
	}
	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 + Float64(abs(x_m) * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (abs(x_m) <= 1e-10)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) * Float64(1.0 / t_0))))))) * Float64(-1.0 / t_1))));
	end
	return 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 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-10], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $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$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000004e-10

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 57.1%

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

      1. distribute-rgt-neg-in 57.1%

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

   8. Simplified 57.1%

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

   9. Taylor expanded in x around 0 97.8%

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

       1. *-commutative 97.8%

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

   11. Simplified 97.8%

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

       1. flip3-+ 97.8%

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

       2. metadata-eval 97.8%

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

       3. metadata-eval 97.8%

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

       4. pow2 97.8%

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

   13. Applied egg-rr 97.8%

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

       1. +-commutative 97.8%

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

       2. cube-prod 97.8%

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

       3. fma-def 97.8%

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

       4. metadata-eval 97.8%

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

       5. cancel-sign-sub-inv 97.8%

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

       6. unpow2 97.8%

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

       7. distribute-rgt-out 97.8%

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

       8. metadata-eval 97.8%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   15. Simplified 97.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 1.00000000000000004e-10 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.5%

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

      3. log1p-udef 99.5%

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

      4. add-exp-log 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-udef 99.5%

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

      7. add-sqr-sqrt 55.1%

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

      8. fabs-sqr 55.1%

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

      9. add-sqr-sqrt 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. fma-udef 99.5%

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

      2. associate--l+ 99.5%

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

      3. metadata-eval 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 99.5%

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

   8. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. expm1-udef 99.5%

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

      3. log1p-udef 99.5%

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

      4. add-exp-log 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-udef 99.5%

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

      7. add-sqr-sqrt 55.1%

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

      8. fabs-sqr 55.1%

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

      9. add-sqr-sqrt 99.5%

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

   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 99.5%

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

       2. associate--l+ 99.5%

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

       3. metadata-eval 99.5%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 98.7%

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

Alternative 5: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left({x_m}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x_m \cdot 1.128386358070218\right) \cdot \left(x_m \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.05)
   (/
    (fma (pow x_m 3.0) 1.436724444676459 1e-27)
    (+
     1e-18
     (* (* x_m 1.128386358070218) (+ (* x_m 1.128386358070218) -1e-9))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.05) {
		tmp = fma(pow(x_m, 3.0), 1.436724444676459, 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) + -1e-9)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.05)
		tmp = Float64(fma((x_m ^ 3.0), 1.436724444676459, 1e-27) / Float64(1e-18 + Float64(Float64(x_m * 1.128386358070218) * Float64(Float64(x_m * 1.128386358070218) + -1e-9))));
	else
		tmp = 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], 0.05], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459 + 1e-27), $MachinePrecision] / N[(1e-18 + N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

   6. Applied egg-rr 56.4%

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

      1. distribute-rgt-neg-in 56.4%

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

   8. Simplified 56.4%

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

   9. Taylor expanded in x around 0 96.3%

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

       1. *-commutative 96.3%

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

   11. Simplified 96.3%

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

       1. flip3-+ 96.3%

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

       2. metadata-eval 96.3%

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

       3. metadata-eval 96.3%

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

       4. pow2 96.3%

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

   13. Applied egg-rr 96.3%

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

       1. +-commutative 96.3%

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

       2. cube-prod 96.3%

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

       3. fma-def 96.3%

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

       4. metadata-eval 96.3%

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

       5. cancel-sign-sub-inv 96.3%

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

       6. unpow2 96.3%

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

       7. distribute-rgt-out 96.3%

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

       8. metadata-eval 96.3%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   15. Simplified 96.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}} \]

   if 0.050000000000000003 < (fabs.f64 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

   6. Applied egg-rr 1.4%

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

      1. distribute-rgt-neg-in 1.4%

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

   8. Simplified 1.4%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 1.436724444676459, 10^{-27}\right)}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 + -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.05:\\ \;\;\;\;10^{-9} + e^{\log \left(x_m \cdot 1.128386358070218\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.05) (+ 1e-9 (exp (log (* x_m 1.128386358070218)))) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.05) {
		tmp = 1e-9 + exp(log((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 (abs(x_m) <= 0.05d0) then
        tmp = 1d-9 + exp(log((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 (Math.abs(x_m) <= 0.05) {
		tmp = 1e-9 + Math.exp(Math.log((x_m * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.05:
		tmp = 1e-9 + math.exp(math.log((x_m * 1.128386358070218)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.05)
		tmp = Float64(1e-9 + exp(log(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 (abs(x_m) <= 0.05)
		tmp = 1e-9 + exp(log((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[N[Abs[x$95$m], $MachinePrecision], 0.05], N[(1e-9 + N[Exp[N[Log[N[(x$95$m * 1.128386358070218), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

   6. Applied egg-rr 56.4%

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

      1. distribute-rgt-neg-in 56.4%

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

   8. Simplified 56.4%

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

   9. Taylor expanded in x around 0 96.3%

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

       1. *-commutative 96.3%

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

   11. Simplified 96.3%

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

       1. add-exp-log 55.9%

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

   13. Applied egg-rr 55.9%

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

   if 0.050000000000000003 < (fabs.f64 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

   6. Applied egg-rr 1.4%

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

      1. distribute-rgt-neg-in 1.4%

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

   8. Simplified 1.4%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;10^{-9} + e^{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.05:\\ \;\;\;\;x_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.05) (+ (* x_m 1.128386358070218) 1e-9) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.05) {
		tmp = (x_m * 1.128386358070218) + 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 (abs(x_m) <= 0.05d0) then
        tmp = (x_m * 1.128386358070218d0) + 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 (Math.abs(x_m) <= 0.05) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.05:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.05)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 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 (abs(x_m) <= 0.05)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.05], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

   6. Applied egg-rr 56.4%

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

      1. distribute-rgt-neg-in 56.4%

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

   8. Simplified 56.4%

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

   9. Taylor expanded in x around 0 96.3%

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

       1. *-commutative 96.3%

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

   11. Simplified 96.3%

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

   if 0.050000000000000003 < (fabs.f64 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

   6. Applied egg-rr 1.4%

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

      1. distribute-rgt-neg-in 1.4%

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

   8. Simplified 1.4%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% 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 71.4%

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

   3. Simplified 71.4%

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

   6. Applied egg-rr 39.6%

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

      1. distribute-rgt-neg-in 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in x around 0 69.1%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 1.2%

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

      1. distribute-rgt-neg-in 1.2%

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

   8. Simplified 1.2%

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

   9. Taylor expanded in x around inf 96.5%

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

4. Final simplification 76.9%

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

Alternative 9: 52.7% 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 79.4%

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

3. Simplified 79.4%

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

6. Applied egg-rr 28.7%

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

   1. distribute-rgt-neg-in 28.7%

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

8. Simplified 28.7%

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

9. Taylor expanded in x around 0 52.6%

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

10. Final simplification 52.6%

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

Reproduce

herbie shell --seed 2024019 
(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)))))))