Jmat.Real.erf

Percentage Accurate: 79.4% → 99.8%

Time: 21.7s

Alternatives: 15

Speedup: 279.5×

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: 79.4% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + -0.254829592}{\mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x_m 3.0))
     (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218))))
   (+
    1.0
    (/
     (+
      (/
       (-
        0.284496736
        (/
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0))
      -0.254829592)
     (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + ((((0.284496736 - ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + -0.254829592) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + -0.254829592) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.284496736 - N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.254829592), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   5. Applied egg-rr 56.9%

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

      1. *-commutative 56.9%

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

      2. distribute-neg-frac 56.9%

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

      3. distribute-neg-in 56.9%

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

      4. metadata-eval 56.9%

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

      5. unsub-neg 56.9%

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

   7. Simplified 56.9%

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

   8. Taylor expanded in x around 0 97.6%

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

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-neg-frac 99.2%

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

      3. distribute-neg-in 99.2%

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

      4. metadata-eval 99.2%

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

      5. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x_m \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{x_m \cdot \left(-x_m\right)}\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 2e-7)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x_m 3.0))
       (+
        (* -0.00011824294398844343 (pow x_m 2.0))
        (* x_m 1.128386358070218))))
     (+
      1.0
      (*
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              t_1
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x_m 0.3275911))))))))))
        (exp (* x_m (- x_m))))
       (/ -1.0 t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))))))))) * exp((x_m * -x_m))) * (-1.0 / t_0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x_m) <= 2d-7) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (((0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / (1.0d0 + (x_m * 0.3275911d0)))))))))) * exp((x_m * -x_m))) * ((-1.0d0) / t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x_m) <= 2e-7) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))))))))) * Math.exp((x_m * -x_m))) * (-1.0 / t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x_m) <= 2e-7:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))))))))) * math.exp((x_m * -x_m))) * (-1.0 / t_0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(x_m * 0.3275911)))))))))) * exp(Float64(x_m * Float64(-x_m)))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x_m) <= 2e-7)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))))))))) * exp((x_m * -x_m))) * (-1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   5. Applied egg-rr 56.9%

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

      1. *-commutative 56.9%

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

      2. distribute-neg-frac 56.9%

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

      3. distribute-neg-in 56.9%

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

      4. metadata-eval 56.9%

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

      5. unsub-neg 56.9%

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

   7. Simplified 56.9%

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

   8. Taylor expanded in x around 0 97.6%

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

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-commutative 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(e^{\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fma-udef 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(e^{\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. add-exp-log 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. add-sqr-sqrt 50.2%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. fabs-sqr 50.2%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      9. add-sqr-sqrt 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.3%

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

      1. fma-udef 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. associate--l+ 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. metadata-eval 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 97.6%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(x_m \cdot -0.3477069720320819 + -0.391746598\right)\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 0.5)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x_m 3.0))
       (+
        (* -0.00011824294398844343 (pow x_m 2.0))
        (* x_m 1.128386358070218))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (* t_1 (+ (* x_m -0.3477069720320819) -0.391746598)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * ((x_m * -0.3477069720320819) + -0.391746598))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x_m * -x_m)) * ((((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((x_m * (-0.3477069720320819d0)) + (-0.391746598d0)))))) * ((-1.0d0) / t_0)) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * ((x_m * -0.3477069720320819) + -0.391746598))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * ((x_m * -0.3477069720320819) + -0.391746598))))) * (-1.0 / t_0)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(Float64(x_m * -0.3477069720320819) + -0.391746598))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * ((x_m * -0.3477069720320819) + -0.391746598))))) * (-1.0 / t_0)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(N[(x$95$m * -0.3477069720320819), $MachinePrecision] + -0.391746598), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-sqr-sqrt 51.6%

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

      7. fabs-sqr 51.6%

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

      8. add-sqr-sqrt 51.6%

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

      9. +-commutative 51.6%

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

      10. fma-udef 51.6%

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

      11. add-sqr-sqrt 51.6%

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

      12. fabs-sqr 51.6%

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

      13. add-sqr-sqrt 51.6%

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

   5. Applied egg-rr 51.6%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. unpow2 99.4%

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

      5. rem-square-sqrt 99.4%

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

      6. metadata-eval 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\color{blue}{-0.3477069720320819} \cdot x + \left({\left(\sqrt{1.061405429}\right)}^{2} - 1.453152027\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. unpow2 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.3477069720320819 \cdot x + \left(\color{blue}{\sqrt{1.061405429} \cdot \sqrt{1.061405429}} - 1.453152027\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. rem-square-sqrt 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.3477069720320819 \cdot x + \left(\color{blue}{1.061405429} - 1.453152027\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      9. metadata-eval 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.3477069720320819 \cdot x + \color{blue}{-0.391746598}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   8. Simplified 99.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(-0.3477069720320819 \cdot x + -0.391746598\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(x \cdot -0.3477069720320819 + -0.391746598\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 + \frac{0.284496736 - \frac{1.029667143}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x_m 3.0))
     (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218))))
   (+
    1.0
    (/
     (+
      -0.254829592
      (/
       (- 0.284496736 (/ 1.029667143 (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0)))
     (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + ((-0.254829592 + ((0.284496736 - (1.029667143 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.254829592 + Float64(Float64(0.284496736 - Float64(1.029667143 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-0.254829592 + N[(N[(0.284496736 - N[(1.029667143 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-udef 100.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 + \frac{0.284496736 - \frac{1.029667143}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\\ \end{array} \]

Alternative 5: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + -0.391746598 \cdot \frac{1}{1 + x_m \cdot 0.3275911}\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 0.5)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x_m 3.0))
       (+
        (* -0.00011824294398844343 (pow x_m 2.0))
        (* x_m 1.128386358070218))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (* -0.391746598 (/ 1.0 (+ 1.0 (* x_m 0.3275911)))))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (-0.391746598 * (1.0 / (1.0 + (x_m * 0.3275911))))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x_m * -x_m)) * ((((-0.284496736d0) + (t_1 * (1.421413741d0 + ((-0.391746598d0) * (1.0d0 / (1.0d0 + (x_m * 0.3275911d0))))))) * ((-1.0d0) / t_0)) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (-0.391746598 * (1.0 / (1.0 + (x_m * 0.3275911))))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (-0.391746598 * (1.0 / (1.0 + (x_m * 0.3275911))))))) * (-1.0 / t_0)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(-0.391746598 * Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + (-0.391746598 * (1.0 / (1.0 + (x_m * 0.3275911))))))) * (-1.0 / t_0)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(-0.391746598 * N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-sqr-sqrt 51.6%

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

      7. fabs-sqr 51.6%

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

      8. add-sqr-sqrt 51.6%

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

      9. +-commutative 51.6%

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

      10. fma-udef 51.6%

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

      11. add-sqr-sqrt 51.6%

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

      12. fabs-sqr 51.6%

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

      13. add-sqr-sqrt 51.6%

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

   5. Applied egg-rr 51.6%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. unpow2 99.4%

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

      2. rem-square-sqrt 99.4%

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

      3. metadata-eval 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{-0.391746598}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   8. Simplified 99.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{-0.391746598}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-commutative 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(e^{\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fma-udef 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(e^{\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. add-exp-log 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. add-sqr-sqrt 51.0%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. fabs-sqr 51.0%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      9. add-sqr-sqrt 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   10. Applied egg-rr 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   11. Step-by-step derivation

       1. fma-udef 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. associate--l+ 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. metadata-eval 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   12. Simplified 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + -0.391746598 \cdot \frac{1}{1 + x \cdot 0.3275911}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 6: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.254829592 \cdot \frac{-1}{e^{{x_m}^{2}} \cdot \left(1 + \left|x_m\right| \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x_m 3.0))
     (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218))))
   (+
    1.0
    (*
     0.254829592
     (/ -1.0 (* (exp (pow x_m 2.0)) (+ 1.0 (* (fabs x_m) 0.3275911))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (0.254829592 * (-1.0 / (exp(pow(x_m, 2.0)) * (1.0 + (fabs(x_m) * 0.3275911)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (0.254829592d0 * ((-1.0d0) / (exp((x_m ** 2.0d0)) * (1.0d0 + (abs(x_m) * 0.3275911d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 + (0.254829592 * (-1.0 / (Math.exp(Math.pow(x_m, 2.0)) * (1.0 + (Math.abs(x_m) * 0.3275911)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 + (0.254829592 * (-1.0 / (math.exp(math.pow(x_m, 2.0)) * (1.0 + (math.fabs(x_m) * 0.3275911)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(0.254829592 * Float64(-1.0 / Float64(exp((x_m ^ 2.0)) * Float64(1.0 + Float64(abs(x_m) * 0.3275911))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 + (0.254829592 * (-1.0 / (exp((x_m ^ 2.0)) * (1.0 + (abs(x_m) * 0.3275911)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.254829592 * N[(-1.0 / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.254829592 \cdot \frac{-1}{e^{{x}^{2}} \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 7: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{-0.7778892405807117}{x_m}}{e^{{x_m}^{2}}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x_m 3.0))
     (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* 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 (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (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 (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (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 (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (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 math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (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 (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + 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 (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (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[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $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 (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. associate-/r* 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{-0.7778892405807117}{x}}{e^{{x}^{2}}}}\\ \end{array} \]

Alternative 8: 99.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x_m 3.0))
     (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* 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 (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (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 (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (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 (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (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 math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (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 (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + 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 (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (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[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $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 (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\\ \end{array} \]

Alternative 9: 99.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* 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 (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (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 (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (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 (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (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 math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (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 (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + 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 (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (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[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $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 (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 95.8%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\\ \end{array} \]

Alternative 10: 99.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x_m}^{3} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+ (* -0.37545125292247583 (pow x_m 3.0)) (* x_m 1.128386358070218)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x_m, 3.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x_m ** 3.0d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x_m, 3.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x_m, 3.0)) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x_m ^ 3.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.37545125292247583 * (x_m ^ 3.0)) + (x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 96.0%

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

   9. Taylor expanded in x around 0 96.0%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-udef 100.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 99.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (+
    1e-9
    (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.5d0) then
        tmp = 1d-9 + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = 1e-9 + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = 1e-9 + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 95.8%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-udef 100.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 99.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5) (fma x_m 1.128386358070218 1e-9) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 58.6%

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

   3. Simplified 58.6%

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

   5. Applied egg-rr 57.1%

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

      1. *-commutative 57.1%

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

      2. distribute-neg-frac 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. metadata-eval 57.1%

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

      5. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in x around 0 95.8%

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

      1. +-commutative 95.8%

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

      2. *-commutative 95.8%

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

      3. fma-def 95.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   10. Simplified 95.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 0.5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-udef 100.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 99.3% accurate, 121.2× 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 71.5%

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

   3. Simplified 71.5%

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

   5. Applied egg-rr 70.5%

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

      1. *-commutative 70.5%

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

      2. distribute-neg-frac 70.5%

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

      3. distribute-neg-in 70.5%

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

      4. metadata-eval 70.5%

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

      5. unsub-neg 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in x around 0 66.2%

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

      1. *-commutative 66.2%

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

   10. Simplified 66.2%

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

   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%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-udef 100.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 98.8%

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

4. Final simplification 74.4%

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

Alternative 14: 97.7% accurate, 279.5× 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 71.3%

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

   3. Simplified 71.3%

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

   5. Applied egg-rr 70.2%

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

      1. *-commutative 70.2%

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

      2. distribute-neg-frac 70.2%

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

      3. distribute-neg-in 70.2%

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

      4. metadata-eval 70.2%

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

      5. unsub-neg 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in x around 0 69.5%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. log1p-udef 100.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 96.3%

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

4. Final simplification 76.4%

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

Alternative 15: 52.7% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 78.7%

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

5. Applied egg-rr 77.9%

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

   1. *-commutative 77.9%

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

   2. distribute-neg-frac 77.9%

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

   3. distribute-neg-in 77.9%

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

   4. metadata-eval 77.9%

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

   5. unsub-neg 77.9%

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

7. Simplified 77.9%

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

8. Taylor expanded in x around 0 54.4%

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

9. Final simplification 54.4%

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

Reproduce

herbie shell --seed 2023321 
(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)))))))