Jmat.Real.erf

Percentage Accurate: 79.3% → 99.9%

Time: 24.4s

Alternatives: 14

Speedup: 142.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x_m \cdot 0.3275911\\ \mathbf{if}\;\left|x_m\right| \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{t_0}^{3}} + 1.421413741 \cdot \frac{1}{t_0}\right) + \left(1.453152027 \cdot \frac{-1}{{t_0}^{2}} - 0.284496736\right)}{1 + \left|x_m\right| \cdot 0.3275911}, \frac{{\left(e^{x_m}\right)}^{\left(-x_m\right)}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911))))
   (if (<= (fabs x_m) 0.0005)
     (+
      (fma
       (pow x_m 3.0)
       -0.37545125292247583
       (* (pow x_m 2.0) -0.00011824294398844343))
      (fma x_m 1.128386358070218 1e-9))
     (fma
      (-
       -0.254829592
       (/
        (+
         (+ (* 1.061405429 (/ 1.0 (pow t_0 3.0))) (* 1.421413741 (/ 1.0 t_0)))
         (- (* 1.453152027 (/ -1.0 (pow t_0 2.0))) 0.284496736))
        (+ 1.0 (* (fabs x_m) 0.3275911))))
      (/ (pow (exp x_m) (- x_m)) (fma 0.3275911 (fabs x_m) 1.0))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double tmp;
	if (fabs(x_m) <= 0.0005) {
		tmp = fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = fma((-0.254829592 - ((((1.061405429 * (1.0 / pow(t_0, 3.0))) + (1.421413741 * (1.0 / t_0))) + ((1.453152027 * (-1.0 / pow(t_0, 2.0))) - 0.284496736)) / (1.0 + (fabs(x_m) * 0.3275911)))), (pow(exp(x_m), -x_m) / fma(0.3275911, fabs(x_m), 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(x_m * 0.3275911))
	tmp = 0.0
	if (abs(x_m) <= 0.0005)
		tmp = Float64(fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9));
	else
		tmp = fma(Float64(-0.254829592 - Float64(Float64(Float64(Float64(1.061405429 * Float64(1.0 / (t_0 ^ 3.0))) + Float64(1.421413741 * Float64(1.0 / t_0))) + Float64(Float64(1.453152027 * Float64(-1.0 / (t_0 ^ 2.0))) - 0.284496736)) / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))), Float64((exp(x_m) ^ Float64(-x_m)) / fma(0.3275911, abs(x_m), 1.0)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0005], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 - N[(N[(N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 * N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.284496736), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-4

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+l+ 96.7%

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

      4. *-commutative 96.7%

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

      5. fma-def 96.7%

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

      6. *-commutative 96.7%

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

      7. *-commutative 96.7%

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

      8. fma-def 96.7%

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

   9. Simplified 96.7%

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

   if 5.0000000000000001e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. expm1-udef 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-prod 100.0%

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

      9. sqr-abs 100.0%

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

      10. sqrt-prod 50.4%

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

      11. add-sqr-sqrt 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. fma-udef 98.7%

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

      2. associate--l+ 98.7%

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

      3. metadata-eval 98.7%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. +-rgt-identity 98.7%

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

      5. *-commutative 98.7%

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

   9. Simplified 98.7%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       2. expm1-udef 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       3. log1p-udef 100.0%

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

       4. add-exp-log 100.0%

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

       5. +-commutative 100.0%

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

       6. fma-udef 100.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. sqrt-prod 100.0%

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

       9. sqr-abs 100.0%

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

       10. sqrt-prod 50.4%

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

       11. add-sqr-sqrt 98.7%

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

   11. Applied egg-rr 98.7%

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

       1. fma-udef 98.7%

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

       2. associate--l+ 98.7%

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

       3. metadata-eval 98.7%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       4. +-rgt-identity 98.7%

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

       5. *-commutative 98.7%

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

   13. Simplified 98.7%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       2. expm1-udef 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       3. log1p-udef 100.0%

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

       4. add-exp-log 100.0%

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

       5. +-commutative 100.0%

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

       6. fma-udef 100.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. sqrt-prod 100.0%

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

       9. sqr-abs 100.0%

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

       10. sqrt-prod 50.4%

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

       11. add-sqr-sqrt 98.7%

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

   15. Applied egg-rr 98.8%

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

       1. fma-udef 98.7%

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

       2. associate--l+ 98.7%

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

       3. metadata-eval 98.7%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       4. +-rgt-identity 98.7%

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

       5. *-commutative 98.7%

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

   17. Simplified 98.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + x \cdot 0.3275911\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + x \cdot 0.3275911}\right) + \left(1.453152027 \cdot \frac{-1}{{\left(1 + x \cdot 0.3275911\right)}^{2}} - 0.284496736\right)}{1 + \left|x\right| \cdot 0.3275911}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.00065:\\ \;\;\;\;\mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00065)
   (+
    (fma
     (pow x_m 3.0)
     -0.37545125292247583
     (* (pow x_m 2.0) -0.00011824294398844343))
    (fma x_m 1.128386358070218 1e-9))
   (-
    1.0
    (/
     (/
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
            (fma x_m 0.3275911 1.0)))
          (fma x_m 0.3275911 1.0)))
        (fma x_m 0.3275911 1.0)))
      (pow (exp x_m) x_m))
     (fma 0.3275911 (fabs x_m) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00065) {
		tmp = fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / pow(exp(x_m), x_m)) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00065)
		tmp = Float64(fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (exp(x_m) ^ x_m)) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00065], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4999999999999997e-4

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

      1. +-commutative 67.0%

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

      2. associate-+r+ 67.0%

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

      3. associate-+l+ 67.0%

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

      4. *-commutative 67.0%

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

      5. fma-def 67.0%

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

      6. *-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. fma-def 67.0%

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

   9. Simplified 67.0%

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

   if 6.4999999999999997e-4 < 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%

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

   6. Applied egg-rr 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00065:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}, \frac{e^{-{x_m}^{2}}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0006)
   (+
    (fma
     (pow x_m 3.0)
     -0.37545125292247583
     (* (pow x_m 2.0) -0.00011824294398844343))
    (fma x_m 1.128386358070218 1e-9))
   (fma
    (-
     -0.254829592
     (/
      (+
       -0.284496736
       (/
        (+
         1.421413741
         (/
          (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
          (fma x_m 0.3275911 1.0)))
        (fma x_m 0.3275911 1.0)))
      (fma x_m 0.3275911 1.0)))
    (/ (exp (- (pow x_m 2.0))) (fma x_m 0.3275911 1.0))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0006) {
		tmp = fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = fma((-0.254829592 - ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))), (exp(-pow(x_m, 2.0)) / fma(x_m, 0.3275911, 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0006)
		tmp = Float64(fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9));
	else
		tmp = fma(Float64(-0.254829592 - Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))), Float64(exp(Float64(-(x_m ^ 2.0))) / fma(x_m, 0.3275911, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0006], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 - N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999947e-4

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

      1. +-commutative 67.0%

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

      2. associate-+r+ 67.0%

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

      3. associate-+l+ 67.0%

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

      4. *-commutative 67.0%

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

      5. fma-def 67.0%

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

      6. *-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. fma-def 67.0%

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

   9. Simplified 67.0%

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

   if 5.99999999999999947e-4 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + x_m \cdot 0.3275911}\\ t_2 := \frac{1}{t_0}\\ \mathbf{if}\;x_m \leq 0.00064:\\ \;\;\;\;\mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t_1 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_2 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right) \cdot e^{x_m \cdot \left(-x_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_1 (/ 1.0 (+ 1.0 (* x_m 0.3275911))))
        (t_2 (/ 1.0 t_0)))
   (if (<= x_m 0.00064)
     (+
      (fma
       (pow x_m 3.0)
       -0.37545125292247583
       (* (pow x_m 2.0) -0.00011824294398844343))
      (fma x_m 1.128386358070218 1e-9))
     (-
      1.0
      (*
       (*
        t_1
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_2
            (+ 1.421413741 (* t_2 (+ -1.453152027 (/ 1.061405429 t_0)))))))))
       (exp (* x_m (- x_m))))))))

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 / (1.0 + (x_m * 0.3275911));
	double t_2 = 1.0 / t_0;
	double tmp;
	if (x_m <= 0.00064) {
		tmp = fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_0))))))))) * exp((x_m * -x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))
	t_2 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 0.00064)
		tmp = Float64(fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9));
	else
		tmp = Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(1.061405429 / t_0))))))))) * exp(Float64(x_m * Float64(-x_m)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00064], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * 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]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.40000000000000052e-4

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

      1. +-commutative 67.0%

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

      2. associate-+r+ 67.0%

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

      3. associate-+l+ 67.0%

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

      4. *-commutative 67.0%

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

      5. fma-def 67.0%

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

      6. *-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. fma-def 67.0%

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

   9. Simplified 67.0%

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

   if 6.40000000000000052e-4 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. expm1-udef 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-prod 100.0%

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

      9. sqr-abs 100.0%

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

      10. sqrt-prod 100.0%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. +-rgt-identity 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. expm1-udef 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-prod 100.0%

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

      9. sqr-abs 100.0%

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

      10. sqrt-prod 100.0%

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

      11. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

       4. +-rgt-identity 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00064:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{\mathsf{fma}\left(x_m, x_m, \mathsf{log1p}\left(x_m \cdot 0.3275911\right)\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (+
    (fma
     (pow x_m 3.0)
     -0.37545125292247583
     (* (pow x_m 2.0) -0.00011824294398844343))
    (fma x_m 1.128386358070218 1e-9))
   (- 1.0 (/ 0.254829592 (exp (fma x_m x_m (log1p (* x_m 0.3275911))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 - (0.254829592 / exp(fma(x_m, x_m, log1p((x_m * 0.3275911)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)) + fma(x_m, 1.128386358070218, 1e-9));
	else
		tmp = Float64(1.0 - Float64(0.254829592 / exp(fma(x_m, x_m, log1p(Float64(x_m * 0.3275911))))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

      1. +-commutative 67.0%

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

      2. associate-+r+ 67.0%

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

      3. associate-+l+ 67.0%

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

      4. *-commutative 67.0%

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

      5. fma-def 67.0%

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

      6. *-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. fma-def 67.0%

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

   9. Simplified 67.0%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.2%

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

      1. add-exp-log 99.2%

         \[\leadsto 1 - \color{blue}{e^{\log \left(\frac{\frac{0.254829592}{e^{{x}^{2}}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}} \]

      2. fma-udef 99.2%

         \[\leadsto 1 - e^{\log \left(\frac{\frac{0.254829592}{e^{{x}^{2}}}}{\color{blue}{x \cdot 0.3275911 + 1}}\right)} \]

      3. +-commutative 99.2%

         \[\leadsto 1 - e^{\log \left(\frac{\frac{0.254829592}{e^{{x}^{2}}}}{\color{blue}{1 + x \cdot 0.3275911}}\right)} \]

      4. log-div 99.2%

         \[\leadsto 1 - e^{\color{blue}{\log \left(\frac{0.254829592}{e^{{x}^{2}}}\right) - \log \left(1 + x \cdot 0.3275911\right)}} \]

      5. log-div 99.2%

         \[\leadsto 1 - e^{\color{blue}{\left(\log 0.254829592 - \log \left(e^{{x}^{2}}\right)\right)} - \log \left(1 + x \cdot 0.3275911\right)} \]

      6. add-log-exp 99.2%

         \[\leadsto 1 - e^{\left(\log 0.254829592 - \color{blue}{{x}^{2}}\right) - \log \left(1 + x \cdot 0.3275911\right)} \]

      7. log1p-udef 99.2%

         \[\leadsto 1 - e^{\left(\log 0.254829592 - {x}^{2}\right) - \color{blue}{\mathsf{log1p}\left(x \cdot 0.3275911\right)}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto 1 - \color{blue}{e^{\left(\log 0.254829592 - {x}^{2}\right) - \mathsf{log1p}\left(x \cdot 0.3275911\right)}} \]
   9. Step-by-step derivation

      1. associate--l- 99.2%

         \[\leadsto 1 - e^{\color{blue}{\log 0.254829592 - \left({x}^{2} + \mathsf{log1p}\left(x \cdot 0.3275911\right)\right)}} \]

      2. exp-diff 99.2%

         \[\leadsto 1 - \color{blue}{\frac{e^{\log 0.254829592}}{e^{{x}^{2} + \mathsf{log1p}\left(x \cdot 0.3275911\right)}}} \]

      3. rem-exp-log 99.2%

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

      4. unpow2 99.2%

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

      5. fma-def 99.2%

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

      6. *-commutative 99.2%

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

   10. Simplified 99.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{\mathsf{fma}\left(x, x, \mathsf{log1p}\left(x \cdot 0.3275911\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;10^{-9} + \left({x_m}^{3} \cdot -0.37545125292247583 + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{\mathsf{fma}\left(x_m, x_m, \mathsf{log1p}\left(x_m \cdot 0.3275911\right)\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (+
    1e-9
    (+
     (* (pow x_m 3.0) -0.37545125292247583)
     (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218))))
   (- 1.0 (/ 0.254829592 (exp (fma x_m x_m (log1p (* x_m 0.3275911))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1e-9 + ((pow(x_m, 3.0) * -0.37545125292247583) + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.254829592 / exp(fma(x_m, x_m, log1p((x_m * 0.3275911)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 3.0) * -0.37545125292247583) + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 - Float64(0.254829592 / exp(fma(x_m, x_m, log1p(Float64(x_m * 0.3275911))))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.2%

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

      1. add-exp-log 99.2%

         \[\leadsto 1 - \color{blue}{e^{\log \left(\frac{\frac{0.254829592}{e^{{x}^{2}}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}} \]

      2. fma-udef 99.2%

         \[\leadsto 1 - e^{\log \left(\frac{\frac{0.254829592}{e^{{x}^{2}}}}{\color{blue}{x \cdot 0.3275911 + 1}}\right)} \]

      3. +-commutative 99.2%

         \[\leadsto 1 - e^{\log \left(\frac{\frac{0.254829592}{e^{{x}^{2}}}}{\color{blue}{1 + x \cdot 0.3275911}}\right)} \]

      4. log-div 99.2%

         \[\leadsto 1 - e^{\color{blue}{\log \left(\frac{0.254829592}{e^{{x}^{2}}}\right) - \log \left(1 + x \cdot 0.3275911\right)}} \]

      5. log-div 99.2%

         \[\leadsto 1 - e^{\color{blue}{\left(\log 0.254829592 - \log \left(e^{{x}^{2}}\right)\right)} - \log \left(1 + x \cdot 0.3275911\right)} \]

      6. add-log-exp 99.2%

         \[\leadsto 1 - e^{\left(\log 0.254829592 - \color{blue}{{x}^{2}}\right) - \log \left(1 + x \cdot 0.3275911\right)} \]

      7. log1p-udef 99.2%

         \[\leadsto 1 - e^{\left(\log 0.254829592 - {x}^{2}\right) - \color{blue}{\mathsf{log1p}\left(x \cdot 0.3275911\right)}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto 1 - \color{blue}{e^{\left(\log 0.254829592 - {x}^{2}\right) - \mathsf{log1p}\left(x \cdot 0.3275911\right)}} \]
   9. Step-by-step derivation

      1. associate--l- 99.2%

         \[\leadsto 1 - e^{\color{blue}{\log 0.254829592 - \left({x}^{2} + \mathsf{log1p}\left(x \cdot 0.3275911\right)\right)}} \]

      2. exp-diff 99.2%

         \[\leadsto 1 - \color{blue}{\frac{e^{\log 0.254829592}}{e^{{x}^{2} + \mathsf{log1p}\left(x \cdot 0.3275911\right)}}} \]

      3. rem-exp-log 99.2%

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

      4. unpow2 99.2%

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

      5. fma-def 99.2%

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

      6. *-commutative 99.2%

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

   10. Simplified 99.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{\mathsf{fma}\left(x, x, \mathsf{log1p}\left(x \cdot 0.3275911\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-/r* 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 74.8%

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

Alternative 8: 99.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.05:\\ \;\;\;\;10^{-9} + \left({x_m}^{3} \cdot -0.37545125292247583 + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05)
   (+
    1e-9
    (+
     (* (pow x_m 3.0) -0.37545125292247583)
     (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218))))
   (- 1.0 (/ 0.7778892405807117 (* x_m (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = 1e-9 + ((pow(x_m, 3.0) * -0.37545125292247583) + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.05d0) then
        tmp = 1d-9 + (((x_m ** 3.0d0) * (-0.37545125292247583d0)) + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x_m * exp((x_m ** 2.0d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = 1e-9 + ((Math.pow(x_m, 3.0) * -0.37545125292247583) + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * Math.exp(Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.05:
		tmp = 1e-9 + ((math.pow(x_m, 3.0) * -0.37545125292247583) + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x_m * math.exp(math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 3.0) * -0.37545125292247583) + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m * exp((x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.05)
		tmp = 1e-9 + (((x_m ^ 3.0) * -0.37545125292247583) + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp((x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.05], N[(1e-9 + N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.0%

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

   if 1.05000000000000004 < 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.85:\\ \;\;\;\;10^{-9} + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.85)
   (+
    1e-9
    (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218)))
   (- 1.0 (/ 0.7778892405807117 (* x_m (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.85) {
		tmp = 1e-9 + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.85d0) then
        tmp = 1d-9 + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x_m * exp((x_m ** 2.0d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.85) {
		tmp = 1e-9 + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * Math.exp(Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.85:
		tmp = 1e-9 + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x_m * math.exp(math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.85)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m * exp((x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.85)
		tmp = 1e-9 + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	else
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp((x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.85], N[(1e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.849999999999999978

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 66.3%

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

   if 0.849999999999999978 < 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;10^{-9} + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;10^{-9} + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+
    1e-9
    (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.88d0) then
        tmp = 1d-9 + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = 1e-9 + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = 1e-9 + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 66.3%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. expm1-udef 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-prod 100.0%

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

      9. sqr-abs 100.0%

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

      10. sqrt-prod 100.0%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. +-rgt-identity 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.3% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;10^{-9} + x_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (+ 1e-9 (* x_m 1.128386358070218)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.88d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 66.1%

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

      1. *-commutative 66.1%

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

   9. Simplified 66.1%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. expm1-udef 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-prod 100.0%

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

      9. sqr-abs 100.0%

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

      10. sqrt-prod 100.0%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. +-rgt-identity 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 142.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.4e-5) 1e-9 0.745170408))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.4e-5) {
		tmp = 1e-9;
	} else {
		tmp = 0.745170408;
	}
	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 <= 2.4d-5) then
        tmp = 1d-9
    else
        tmp = 0.745170408d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.4e-5) {
		tmp = 1e-9;
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.4e-5:
		tmp = 1e-9
	else:
		tmp = 0.745170408
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.4e-5)
		tmp = 1e-9;
	else
		tmp = 0.745170408;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.4e-5)
		tmp = 1e-9;
	else
		tmp = 0.745170408;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.4e-5], 1e-9, 0.745170408]

Derivation

1. Split input into 2 regimes

2. if x < 2.4000000000000001e-5

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.8%

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

   if 2.4000000000000001e-5 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 97.8%

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

   7. Taylor expanded in x around 0 20.2%

      \[\leadsto \color{blue}{0.745170408} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.6% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in x around 0 67.8%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      2. expm1-udef 99.7%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      3. log1p-udef 99.7%

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

      4. add-exp-log 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-udef 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. sqrt-prod 99.7%

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

      9. sqr-abs 99.7%

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

      10. sqrt-prod 99.7%

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

      11. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. associate--l+ 99.7%

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

      3. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. +-rgt-identity 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 97.8%

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

4. Final simplification 75.2%

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

Alternative 14: 52.8% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

6. Applied egg-rr 76.9%

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

7. Taylor expanded in x around 0 53.8%

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

8. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2024021 
(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)))))))