Jmat.Real.erf

Percentage Accurate: 78.8% → 99.4%

Time: 29.1s

Alternatives: 11

Speedup: 142.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ t_1 := e^{{x\_m}^{2}}\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.284496736}{{t\_0}^{2} \cdot t\_1} + \left(1 + \frac{\frac{1.453152027}{t\_1}}{{t\_0}^{4}}\right)\right) - \frac{0.254829592}{t\_0 \cdot t\_1}\right) - \left(\frac{1.061405429}{t\_1 \cdot {t\_0}^{5}} + \frac{1.421413741}{t\_1 \cdot {t\_0}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x_m) 1.0)) (t_1 (exp (pow x_m 2.0))))
   (if (<= (fabs x_m) 2e-13)
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (- (* x_m 1.128386358070218) 1e-9))))
     (-
      (-
       (+
        (/ 0.284496736 (* (pow t_0 2.0) t_1))
        (+ 1.0 (/ (/ 1.453152027 t_1) (pow t_0 4.0))))
       (/ 0.254829592 (* t_0 t_1)))
      (+
       (/ 1.061405429 (* t_1 (pow t_0 5.0)))
       (/ 1.421413741 (* t_1 (pow t_0 3.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(0.3275911, fabs(x_m), 1.0);
	double t_1 = exp(pow(x_m, 2.0));
	double tmp;
	if (fabs(x_m) <= 2e-13) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) - 1e-9)));
	} else {
		tmp = (((0.284496736 / (pow(t_0, 2.0) * t_1)) + (1.0 + ((1.453152027 / t_1) / pow(t_0, 4.0)))) - (0.254829592 / (t_0 * t_1))) - ((1.061405429 / (t_1 * pow(t_0, 5.0))) + (1.421413741 / (t_1 * pow(t_0, 3.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, abs(x_m), 1.0)
	t_1 = exp((x_m ^ 2.0))
	tmp = 0.0
	if (abs(x_m) <= 2e-13)
		tmp = Float64(Float64(Float64((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(Float64(Float64(0.284496736 / Float64((t_0 ^ 2.0) * t_1)) + Float64(1.0 + Float64(Float64(1.453152027 / t_1) / (t_0 ^ 4.0)))) - Float64(0.254829592 / Float64(t_0 * t_1))) - Float64(Float64(1.061405429 / Float64(t_1 * (t_0 ^ 5.0))) + Float64(1.421413741 / Float64(t_1 * (t_0 ^ 3.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-13], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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[(N[(N[(0.284496736 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(1.453152027 / t$95$1), $MachinePrecision] / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.254829592 / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.061405429 / N[(t$95$1 * N[Power[t$95$0, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 / N[(t$95$1 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.0000000000000001e-13

   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%

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

   6. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 55.3%

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

      2. sub-neg 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in x around 0 99.1%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.1%

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

       1. flip3-+ 99.1%

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

       2. metadata-eval 99.1%

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

       3. unpow-prod-down 99.1%

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

       4. metadata-eval 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow2 99.1%

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

   13. Applied egg-rr 99.1%

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

       1. +-commutative 99.1%

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

       2. unpow2 99.1%

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

       3. distribute-rgt-out-- 99.1%

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

   15. Simplified 99.1%

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

   if 2.0000000000000001e-13 < (fabs.f64 x)

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(1 + \left(0.284496736 \cdot \frac{1}{e^{{x}^{2}} \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + 1.453152027 \cdot \frac{1}{e^{{x}^{2}} \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right)\right) - \left(0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \left(1.061405429 \cdot \frac{1}{e^{{x}^{2}} \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{1}{e^{{x}^{2}} \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2} \cdot e^{{x}^{2}}} + \left(\frac{\frac{1.453152027}{e^{{x}^{2}}}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}} + 1\right)\right) - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{{x}^{2}}}\right) - \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{5} \cdot e^{{x}^{2}}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3} \cdot e^{{x}^{2}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2} \cdot e^{{x}^{2}}} + \left(1 + \frac{\frac{1.453152027}{e^{{x}^{2}}}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right)\right) - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{{x}^{2}}}\right) - \left(\frac{1.061405429}{e^{{x}^{2}} \cdot {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{5}} + \frac{1.421413741}{e^{{x}^{2}} \cdot {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{e^{{x\_m}^{2}} \cdot \mathsf{fma}\left(0.3275911, x\_m, 1\right)}\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_0}^{2}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               (+
                2.421413741
                (/
                 (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                 (fma 0.3275911 x_m 1.0)))
               -1.0)
              (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (* (exp (pow x_m 2.0)) (fma 0.3275911 x_m 1.0)))))
   (if (<= (fabs x_m) 2e-13)
     (/
      (+ (* (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 2.0)) (- t_0 -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.254829592 + ((-0.284496736 + (((2.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) + -1.0) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (exp(pow(x_m, 2.0)) * fma(0.3275911, x_m, 1.0));
	double tmp;
	if (fabs(x_m) <= 2e-13) {
		tmp = ((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, 2.0)) / (t_0 - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(Float64(2.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) + -1.0) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(exp((x_m ^ 2.0)) * fma(0.3275911, x_m, 1.0)))
	tmp = 0.0
	if (abs(x_m) <= 2e-13)
		tmp = Float64(Float64(Float64((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 - (t_0 ^ 2.0)) / Float64(t_0 - -1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(N[(2.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-13], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.0000000000000001e-13

   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%

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

   6. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 55.3%

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

      2. sub-neg 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in x around 0 99.1%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.1%

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

       1. flip3-+ 99.1%

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

       2. metadata-eval 99.1%

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

       3. unpow-prod-down 99.1%

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

       4. metadata-eval 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow2 99.1%

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

   13. Applied egg-rr 99.1%

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

       1. +-commutative 99.1%

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

       2. unpow2 99.1%

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

       3. distribute-rgt-out-- 99.1%

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

   15. Simplified 99.1%

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

   if 2.0000000000000001e-13 < (fabs.f64 x)

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

      1. add0 99.7%

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

   6. Applied egg-rr 97.7%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + 0\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
   7. Step-by-step derivation

      1. add0 97.7%

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

   8. Simplified 97.7%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-undefine 97.7%

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

   10. Applied egg-rr 97.7%

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

       1. sub-neg 97.7%

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

       2. log1p-undefine 97.7%

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

       3. rem-exp-log 97.7%

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

       4. associate-+r+ 97.7%

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

       5. metadata-eval 97.7%

          \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(\color{blue}{2.421413741} + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \left(-1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

       6. metadata-eval 97.7%

          \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \color{blue}{-1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

   12. Simplified 97.7%

       \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

   14. Applied egg-rr 97.7%

       \[\leadsto \color{blue}{\frac{-\left(1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)}{-\left(1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}} \]
   15. Step-by-step derivation

   16. Simplified 97.7%

       \[\leadsto \color{blue}{\frac{-\left(1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}\right)}{-1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{-e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + 1.061405429 \cdot t\_0\right)\right)\right)\right)\right) \cdot e^{x\_m \cdot \left(-x\_m\right)} \leq 0.9995:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{-1 + e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right) \cdot e^{x\_m \cdot x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* 1.061405429 t_0)))))))))
         (exp (* x_m (- x_m))))
        0.9995)
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            -1.0
            (exp
             (log1p
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0))))))
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (* (fma 0.3275911 (fabs x_m) 1.0) (exp (* x_m x_m)))))
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (- (* x_m 1.128386358070218) 1e-9)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 * t_0))))))))) * exp((x_m * -x_m))) <= 0.9995) {
		tmp = 1.0 - ((0.254829592 + ((-0.284496736 + ((-1.0 + exp(log1p((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, fabs(x_m), 1.0) * exp((x_m * x_m))));
	} else {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) - 1e-9)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (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(1.061405429 * t_0))))))))) * exp(Float64(x_m * Float64(-x_m)))) <= 0.9995)
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(-1.0 + exp(log1p(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, abs(x_m), 1.0) * exp(Float64(x_m * x_m)))));
	else
		tmp = Float64(Float64(Float64((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))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(1.061405429 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.9995], N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(-1.0 + N[Exp[N[Log[1 + N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.99950000000000006

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

      1. add0 99.7%

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

   6. Applied egg-rr 97.7%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + 0\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
   7. Step-by-step derivation

      1. add0 97.7%

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

   8. Simplified 97.7%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-undefine 97.7%

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

   10. Applied egg-rr 97.7%

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

   if 0.99950000000000006 < (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   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%

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

   6. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 55.3%

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

      2. sub-neg 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in x around 0 99.1%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.1%

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

       1. flip3-+ 99.1%

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

       2. metadata-eval 99.1%

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

       3. unpow-prod-down 99.1%

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

       4. metadata-eval 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow2 99.1%

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

   13. Applied egg-rr 99.1%

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

       1. +-commutative 99.1%

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

       2. unpow2 99.1%

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

       3. distribute-rgt-out-- 99.1%

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

   15. Simplified 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + 1.061405429 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)} \leq 0.9995:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{-1 + e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + 1.061405429 \cdot t\_0\right)\right)\right)\right)\right) \cdot e^{x\_m \cdot \left(-x\_m\right)} \leq 0.9995:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{e^{{x\_m}^{2}} \cdot \mathsf{fma}\left(0.3275911, x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* 1.061405429 t_0)))))))))
         (exp (* x_m (- x_m))))
        0.9995)
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            (+
             2.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            -1.0)
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (* (exp (pow x_m 2.0)) (fma 0.3275911 x_m 1.0))))
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (- (* x_m 1.128386358070218) 1e-9)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 * t_0))))))))) * exp((x_m * -x_m))) <= 0.9995) {
		tmp = 1.0 - ((0.254829592 + ((-0.284496736 + (((2.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) + -1.0) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (exp(pow(x_m, 2.0)) * fma(0.3275911, x_m, 1.0)));
	} else {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((x_m * 1.128386358070218) - 1e-9)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (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(1.061405429 * t_0))))))))) * exp(Float64(x_m * Float64(-x_m)))) <= 0.9995)
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(Float64(2.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) + -1.0) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(exp((x_m ^ 2.0)) * fma(0.3275911, x_m, 1.0))));
	else
		tmp = Float64(Float64(Float64((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))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(1.061405429 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.9995], N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(N[(2.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.99950000000000006

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

      1. add0 99.7%

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

   6. Applied egg-rr 97.7%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + 0\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
   7. Step-by-step derivation

      1. add0 97.7%

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

   8. Simplified 97.7%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-undefine 97.7%

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

   10. Applied egg-rr 97.7%

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

       1. sub-neg 97.7%

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

       2. log1p-undefine 97.7%

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

       3. rem-exp-log 97.7%

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

       4. associate-+r+ 97.7%

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

       5. metadata-eval 97.7%

          \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(\color{blue}{2.421413741} + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \left(-1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

       6. metadata-eval 97.7%

          \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \color{blue}{-1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

   12. Simplified 97.7%

       \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
   13. Step-by-step derivation

       1. sub-neg 97.7%

          \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)} \]

       2. associate-+l+ 97.7%

          \[\leadsto 1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \]

       3. *-commutative 97.7%

          \[\leadsto 1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}}\right) \]

       4. add-sqr-sqrt 57.3%

          \[\leadsto 1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) \cdot e^{x \cdot x}}\right) \]

       5. fabs-sqr 57.3%

          \[\leadsto 1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) \cdot e^{x \cdot x}}\right) \]

       6. add-sqr-sqrt 97.7%

          \[\leadsto 1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) \cdot e^{x \cdot x}}\right) \]

   14. Applied egg-rr 97.7%

       \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)} \]
   15. Step-by-step derivation

       1. sub-neg 97.7%

          \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{2.421413741 + \left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

       2. associate-+r+ 97.7%

          \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} \]

       3. *-commutative 97.7%

          \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\color{blue}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   16. Simplified 97.7%

       \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}} \]

   if 0.99950000000000006 < (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   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%

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

   6. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 55.3%

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

      2. sub-neg 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in x around 0 99.1%

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

       1. *-commutative 99.1%

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

   11. Simplified 99.1%

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

       1. flip3-+ 99.1%

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

       2. metadata-eval 99.1%

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

       3. unpow-prod-down 99.1%

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

       4. metadata-eval 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow2 99.1%

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

   13. Applied egg-rr 99.1%

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

       1. +-commutative 99.1%

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

       2. unpow2 99.1%

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

       3. distribute-rgt-out-- 99.1%

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

   15. Simplified 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + 1.061405429 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)} \leq 0.9995:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + t\_1 \cdot \left(-0.284496736 + \left(\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + -1\right) \cdot t\_1\right)\right) \cdot e^{x\_m \cdot \left(-x\_m\right)}\right) \cdot \frac{-1}{t\_0}\\ \end{array} \end{array} \]

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 t_0)))
   (if (<= x_m 1.8e-6)
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       1e-18
       (* (* x_m 1.128386358070218) (- (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            (+
             (+
              2.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
               (fma 0.3275911 x_m 1.0)))
             -1.0)
            t_1))))
        (exp (* x_m (- x_m))))
       (/ -1.0 t_0))))))

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 / t_0;
	double tmp;
	if (x_m <= 1.8e-6) {
		tmp = ((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 + (((0.254829592 + (t_1 * (-0.284496736 + (((2.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) + -1.0) * t_1)))) * exp((x_m * -x_m))) * (-1.0 / t_0));
	}
	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 / t_0)
	tmp = 0.0
	if (x_m <= 1.8e-6)
		tmp = Float64(Float64(Float64((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(Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(Float64(Float64(2.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) + -1.0) * t_1)))) * exp(Float64(x_m * Float64(-x_m)))) * Float64(-1.0 / t_0)));
	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 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.8e-6], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(N[(N[(2.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.79999999999999992e-6

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 39.2%

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

      2. sub-neg 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in x around 0 68.8%

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

       1. *-commutative 68.8%

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

   11. Simplified 68.8%

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

       1. flip3-+ 68.6%

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

       2. metadata-eval 68.6%

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

       3. unpow-prod-down 68.6%

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

       4. metadata-eval 68.6%

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

       5. metadata-eval 68.6%

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

       6. pow2 68.6%

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

   13. Applied egg-rr 68.6%

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

       1. +-commutative 68.6%

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

       2. unpow2 68.6%

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

       3. distribute-rgt-out-- 68.6%

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

   15. Simplified 68.6%

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

   if 1.79999999999999992e-6 < 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%

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

      1. associate-*l/ 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. expm1-log1p-u 100.0%

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

      8. expm1-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-undefine 100.0%

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

      3. rem-exp-log 100.0%

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

      4. associate-+r+ 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(\color{blue}{2.421413741} + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \left(-1\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \color{blue}{-1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   8. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1\right)}\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \left(\left(2.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + -1\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)\right) \cdot e^{x \cdot \left(-x\right)}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(0.254829592 + t\_1 \cdot \left(-0.284496736 + t\_1 \cdot \left(1.421413741 + t\_1 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x\_m \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{t\_0}\\ \end{array} \end{array} \]

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 t_0)))
   (if (<= x_m 1.15e-6)
     (/
      (+ (* (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_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              t_1
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x_m 0.3275911)))))))))))
       (/ -1.0 t_0))))))

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 / t_0;
	double tmp;
	if (x_m <= 1.15e-6) {
		tmp = ((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_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911))))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (x_m <= 1.15d-6) then
        tmp = (((x_m ** 3.0d0) * 1.436724444676459d0) + 1d-27) / (1d-18 + ((x_m * 1.128386358070218d0) * ((x_m * 1.128386358070218d0) - 1d-9)))
    else
        tmp = 1.0d0 + ((exp((x_m * -x_m)) * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / (1.0d0 + (x_m * 0.3275911d0))))))))))) * ((-1.0d0) / t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 1.15e-6) {
		tmp = ((Math.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 + ((Math.exp((x_m * -x_m)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911))))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if x_m <= 1.15e-6:
		tmp = ((math.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 + ((math.exp((x_m * -x_m)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911))))))))))) * (-1.0 / t_0))
	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 / t_0)
	tmp = 0.0
	if (x_m <= 1.15e-6)
		tmp = Float64(Float64(Float64((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(Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(0.254829592 + Float64(t_1 * 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 / t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x_m <= 1.15e-6)
		tmp = (((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_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911))))))))))) * (-1.0 / t_0));
	end
	tmp_2 = 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 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.15e-6], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e-6

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 39.2%

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

      2. sub-neg 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in x around 0 68.8%

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

       1. *-commutative 68.8%

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

   11. Simplified 68.8%

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

       1. flip3-+ 68.6%

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

       2. metadata-eval 68.6%

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

       3. unpow-prod-down 68.6%

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

       4. metadata-eval 68.6%

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

       5. metadata-eval 68.6%

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

       6. pow2 68.6%

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

   13. Applied egg-rr 68.6%

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

       1. +-commutative 68.6%

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

       2. unpow2 68.6%

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

       3. distribute-rgt-out-- 68.6%

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

   15. Simplified 68.6%

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

   if 1.15e-6 < 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%

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

      1. add0 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fabs-sqr 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x\_m \cdot 1.128386358070218\right) \cdot \left(x\_m \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9)
   (/
    (+ (* (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 (x_m <= 0.9) {
		tmp = ((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;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = (((x_m ** 3.0d0) * 1.436724444676459d0) + 1d-27) / (1d-18 + ((x_m * 1.128386358070218d0) * ((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 (x_m <= 0.9) {
		tmp = ((Math.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;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = ((math.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 (x_m <= 0.9)
		tmp = Float64(Float64(Float64((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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = (((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / (1e-18 + ((x_m * 1.128386358070218) * ((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[x$95$m, 0.9], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 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 x < 0.900000000000000022

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 39.2%

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

      2. sub-neg 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in x around 0 68.8%

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

       1. *-commutative 68.8%

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

   11. Simplified 68.8%

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

       1. flip3-+ 68.6%

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

       2. metadata-eval 68.6%

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

       3. unpow-prod-down 68.6%

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

       4. metadata-eval 68.6%

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

       5. metadata-eval 68.6%

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

       6. pow2 68.6%

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

   13. Applied egg-rr 68.6%

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

       1. +-commutative 68.6%

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

       2. unpow2 68.6%

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

       3. distribute-rgt-out-- 68.6%

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

   15. Simplified 68.6%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

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

      2. sub-neg 0.0%

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

   8. Simplified 0.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{{x}^{3} \cdot 1.436724444676459 + 10^{-27}}{10^{-18} + \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218 - 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;\frac{10^{-18} - {x\_m}^{2} \cdot 1.2732557730789702}{10^{-9} + x\_m \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9)
   (/
    (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))
    (+ 1e-9 (* x_m -1.128386358070218)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)) / (1e-9 + (x_m * -1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = (1d-18 - ((x_m ** 2.0d0) * 1.2732557730789702d0)) / (1d-9 + (x_m * (-1.128386358070218d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (1e-18 - (Math.pow(x_m, 2.0) * 1.2732557730789702)) / (1e-9 + (x_m * -1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = (1e-18 - (math.pow(x_m, 2.0) * 1.2732557730789702)) / (1e-9 + (x_m * -1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x_m * -1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = (1e-18 - ((x_m ^ 2.0) * 1.2732557730789702)) / (1e-9 + (x_m * -1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 39.2%

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

      2. sub-neg 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in x around 0 68.8%

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

       1. *-commutative 68.8%

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

   11. Simplified 68.8%

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

       1. flip-+ 68.7%

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

       2. metadata-eval 68.7%

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

       3. pow2 68.7%

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

   13. Applied egg-rr 68.7%

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

       1. unpow2 68.7%

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

       2. swap-sqr 68.7%

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

       3. unpow2 68.7%

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

       4. metadata-eval 68.7%

          \[\leadsto \frac{10^{-18} - {x}^{2} \cdot \color{blue}{1.2732557730789702}}{10^{-9} - x \cdot 1.128386358070218} \]

       5. sub-neg 68.7%

          \[\leadsto \frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{\color{blue}{10^{-9} + \left(-x \cdot 1.128386358070218\right)}} \]

       6. distribute-rgt-neg-in 68.7%

          \[\leadsto \frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + \color{blue}{x \cdot \left(-1.128386358070218\right)}} \]

       7. metadata-eval 68.7%

          \[\leadsto \frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot \color{blue}{-1.128386358070218}} \]

   15. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

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

      2. sub-neg 0.0%

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

   8. Simplified 0.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 85.5× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		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 (x_m <= 0.9d0) 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 (x_m <= 0.9) {
		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 x_m <= 0.9:
		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 (x_m <= 0.9)
		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 (x_m <= 0.9)
		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[x$95$m, 0.9], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 39.2%

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

      2. sub-neg 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in x around 0 68.8%

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

       1. *-commutative 68.8%

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

   11. Simplified 68.8%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

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

      2. sub-neg 0.0%

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

   8. Simplified 0.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 77.9%

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

Alternative 10: 97.7% 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 70.5%

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

   3. Simplified 70.5%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 39.2%

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

      2. sub-neg 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in x around 0 71.5%

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

   if 2.79999999999999996e-5 < 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%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

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

      2. sub-neg 0.0%

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

   8. Simplified 0.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 79.8%

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

Alternative 11: 53.9% 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 79.2%

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

3. Simplified 79.2%

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

6. Applied egg-rr 28.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-undefine 27.7%

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

   2. sub-neg 27.7%

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

8. Simplified 27.9%

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

9. Taylor expanded in x around 0 53.8%

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

10. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2024046 
(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)))))))