Jmat.Real.erf

Percentage Accurate: 79.3% → 99.3%

Time: 14.2s

Alternatives: 8

Speedup: 121.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_0, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (sqrt
          (fma
           -0.00011824294398844343
           (* x x)
           (fma -0.37545125292247583 (pow x 3.0) (* x 1.128386358070218))))))
   (if (<= (fabs x) 0.0001)
     (fma t_0 t_0 1e-9)
     (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = sqrt(fma(-0.00011824294398844343, (x * x), fma(-0.37545125292247583, pow(x, 3.0), (x * 1.128386358070218))));
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = fma(t_0, t_0, 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = sqrt(fma(-0.00011824294398844343, Float64(x * x), fma(-0.37545125292247583, (x ^ 3.0), Float64(x * 1.128386358070218))))
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = fma(t_0, t_0, 1e-9);
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[Sqrt[N[(-0.00011824294398844343 * N[(x * x), $MachinePrecision] + N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(t$95$0 * t$95$0 + 1e-9), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 58.0%

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

   3. Applied egg-rr 57.7%

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

      1. distribute-neg-frac 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. add-cbrt-cube 97.6%

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

   8. Applied egg-rr 97.6%

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

      1. add-cbrt-cube 99.0%

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

      2. +-commutative 99.0%

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

      3. add-sqr-sqrt 53.5%

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

      4. fma-def 53.5%

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

      5. *-commutative 53.5%

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

      6. *-commutative 53.5%

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

   10. Applied egg-rr 53.5%

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

   if 1.00000000000000005e-4 < (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. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)}, \sqrt{\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)}, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 2: 99.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{\left(x \cdot x\right) \cdot 1.2732557730789702 + -1 \cdot 10^{-18}}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (+
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583)))
    (/
     (+ (* (* x x) 1.2732557730789702) -1e-18)
     (fma 1.128386358070218 x -1e-9)))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((((x * x) * 1.2732557730789702) + -1e-18) / fma(1.128386358070218, x, -1e-9));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))) + Float64(Float64(Float64(Float64(x * x) * 1.2732557730789702) + -1e-18) / fma(1.128386358070218, x, -1e-9)));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x * x), $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + -1e-18), $MachinePrecision] / N[(1.128386358070218 * x + -1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 58.0%

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

   3. Applied egg-rr 57.7%

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

      1. distribute-neg-frac 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. *-un-lft-identity 99.0%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      4. fma-def 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      5. *-commutative 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 99.0%

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

      2. +-commutative 99.0%

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

      3. fma-udef 99.0%

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

      4. fma-udef 99.0%

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

      5. associate-+r+ 99.0%

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

      6. unpow2 99.0%

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

      7. associate-+l+ 99.0%

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

      8. *-commutative 99.0%

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

      9. *-commutative 99.0%

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

      10. unpow3 99.0%

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

      11. unpow2 99.0%

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

      12. associate-*l* 99.0%

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

      13. distribute-lft-out 99.0%

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

      14. unpow2 99.0%

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

      15. *-commutative 99.0%

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

      16. fma-def 99.0%

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

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.0%

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

       2. *-commutative 99.0%

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

       3. flip-+ 99.0%

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

       4. pow2 99.0%

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

       5. *-commutative 99.0%

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

       6. metadata-eval 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{{\left(1.128386358070218 \cdot x\right)}^{2} - \color{blue}{10^{-18}}}{x \cdot 1.128386358070218 - 10^{-9}} \]

       7. *-commutative 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{{\left(1.128386358070218 \cdot x\right)}^{2} - 10^{-18}}{\color{blue}{1.128386358070218 \cdot x} - 10^{-9}} \]

       8. fma-neg 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{{\left(1.128386358070218 \cdot x\right)}^{2} - 10^{-18}}{\color{blue}{\mathsf{fma}\left(1.128386358070218, x, -10^{-9}\right)}} \]

       9. metadata-eval 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{{\left(1.128386358070218 \cdot x\right)}^{2} - 10^{-18}}{\mathsf{fma}\left(1.128386358070218, x, \color{blue}{-1 \cdot 10^{-9}}\right)} \]

   12. Applied egg-rr 99.0%

       \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \color{blue}{\frac{{\left(1.128386358070218 \cdot x\right)}^{2} - 10^{-18}}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)}} \]
   13. Step-by-step derivation

       1. sub-neg 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{\color{blue}{{\left(1.128386358070218 \cdot x\right)}^{2} + \left(-10^{-18}\right)}}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)} \]

       2. unpow2 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{\color{blue}{\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right)} + \left(-10^{-18}\right)}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)} \]

       3. swap-sqr 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{\color{blue}{\left(1.128386358070218 \cdot 1.128386358070218\right) \cdot \left(x \cdot x\right)} + \left(-10^{-18}\right)}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)} \]

       4. metadata-eval 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{\color{blue}{1.2732557730789702} \cdot \left(x \cdot x\right) + \left(-10^{-18}\right)}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)} \]

       5. metadata-eval 99.0%

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{1.2732557730789702 \cdot \left(x \cdot x\right) + \color{blue}{-1 \cdot 10^{-18}}}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)} \]

   14. Simplified 99.0%

       \[\leadsto \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \color{blue}{\frac{1.2732557730789702 \cdot \left(x \cdot x\right) + -1 \cdot 10^{-18}}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)}} \]

   if 1.00000000000000005e-4 < (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. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \frac{\left(x \cdot x\right) \cdot 1.2732557730789702 + -1 \cdot 10^{-18}}{\mathsf{fma}\left(1.128386358070218, x, -1 \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 3: 99.7% accurate, 7.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.05)
   (+
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583)))
    (+ (* x 1.128386358070218) 1e-9))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.05d0) then
        tmp = ((x * x) * ((-0.00011824294398844343d0) + (x * (-0.37545125292247583d0)))) + ((x * 1.128386358070218d0) + 1d-9)
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x * exp((x * x))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.05], N[(N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 72.4%

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

   3. Applied egg-rr 72.2%

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

      1. distribute-neg-frac 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in x around 0 66.0%

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

      1. *-un-lft-identity 66.0%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      4. fma-def 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      5. *-commutative 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   8. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 66.0%

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

      2. +-commutative 66.0%

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

      3. fma-udef 66.0%

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

      4. fma-udef 66.0%

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

      5. associate-+r+ 66.0%

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

      6. unpow2 66.0%

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

      7. associate-+l+ 66.0%

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

      8. *-commutative 66.0%

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

      9. *-commutative 66.0%

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

      10. unpow3 66.0%

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

      11. unpow2 66.0%

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

      12. associate-*l* 66.0%

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

      13. distribute-lft-out 66.5%

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

      14. unpow2 66.5%

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

      15. *-commutative 66.5%

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

      16. fma-def 66.5%

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

   10. Simplified 66.5%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]
   11. Step-by-step derivation

       1. fma-udef 66.5%

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

   12. Applied egg-rr 66.5%

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

   if 1.05000000000000004 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 74.8%

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

Alternative 4: 99.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \left(x \cdot 1.128386358070218 + 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x + {x}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.15)
   (+
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583)))
    (+ (* x 1.128386358070218) 1e-9))
   (- 1.0 (/ 0.7778892405807117 (+ x (pow x 3.0))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x + pow(x, 3.0)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.15d0) then
        tmp = ((x * x) * ((-0.00011824294398844343d0) + (x * (-0.37545125292247583d0)))) + ((x * 1.128386358070218d0) + 1d-9)
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x + (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x + Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.15:
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9)
	else:
		tmp = 1.0 - (0.7778892405807117 / (x + math.pow(x, 3.0)))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.15)
		tmp = Float64(Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))) + Float64(Float64(x * 1.128386358070218) + 1e-9));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x + (x ^ 3.0))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.15)
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9);
	else
		tmp = 1.0 - (0.7778892405807117 / (x + (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.15], N[(N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 72.4%

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

   3. Applied egg-rr 72.2%

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

      1. distribute-neg-frac 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in x around 0 66.0%

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

      1. *-un-lft-identity 66.0%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      4. fma-def 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      5. *-commutative 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   8. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 66.0%

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

      2. +-commutative 66.0%

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

      3. fma-udef 66.0%

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

      4. fma-udef 66.0%

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

      5. associate-+r+ 66.0%

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

      6. unpow2 66.0%

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

      7. associate-+l+ 66.0%

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

      8. *-commutative 66.0%

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

      9. *-commutative 66.0%

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

      10. unpow3 66.0%

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

      11. unpow2 66.0%

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

      12. associate-*l* 66.0%

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

      13. distribute-lft-out 66.5%

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

      14. unpow2 66.5%

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

      15. *-commutative 66.5%

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

      16. fma-def 66.5%

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

   10. Simplified 66.5%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]
   11. Step-by-step derivation

       1. fma-udef 66.5%

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

   12. Applied egg-rr 66.5%

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

   if 1.1499999999999999 < 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. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \left(x \cdot 1.128386358070218 + 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x + {x}^{3}}\\ \end{array} \]

Alternative 5: 98.7% accurate, 50.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \left(x \cdot 1.128386358070218 + 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.5)
   (+
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583)))
    (+ (* x 1.128386358070218) 1e-9))
   (- 1.0 (/ 0.7778892405807117 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = ((x * x) * ((-0.00011824294398844343d0) + (x * (-0.37545125292247583d0)))) + ((x * 1.128386358070218d0) + 1d-9)
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9)
	else:
		tmp = 1.0 - (0.7778892405807117 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))) + Float64(Float64(x * 1.128386358070218) + 1e-9));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583))) + ((x * 1.128386358070218) + 1e-9);
	else
		tmp = 1.0 - (0.7778892405807117 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.5], N[(N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 72.4%

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

   3. Applied egg-rr 72.2%

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

      1. distribute-neg-frac 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in x around 0 66.0%

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

      1. *-un-lft-identity 66.0%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      4. fma-def 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      5. *-commutative 66.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   8. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 66.0%

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

      2. +-commutative 66.0%

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

      3. fma-udef 66.0%

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

      4. fma-udef 66.0%

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

      5. associate-+r+ 66.0%

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

      6. unpow2 66.0%

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

      7. associate-+l+ 66.0%

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

      8. *-commutative 66.0%

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

      9. *-commutative 66.0%

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

      10. unpow3 66.0%

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

      11. unpow2 66.0%

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

      12. associate-*l* 66.0%

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

      13. distribute-lft-out 66.5%

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

      14. unpow2 66.5%

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

      15. *-commutative 66.5%

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

      16. fma-def 66.5%

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

   10. Simplified 66.5%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]
   11. Step-by-step derivation

       1. fma-udef 66.5%

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

   12. Applied egg-rr 66.5%

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

   if 1.5 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 99.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right) + \left(x \cdot 1.128386358070218 + 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \]

Alternative 6: 98.5% accurate, 65.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\left(x \cdot 1.128386358070218 + 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.65)
   (+ (+ (* x 1.128386358070218) 1e-9) (* x (* x -0.00011824294398844343)))
   (- 1.0 (/ 0.7778892405807117 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = ((x * 1.128386358070218) + 1e-9) + (x * (x * -0.00011824294398844343));
	} else {
		tmp = 1.0 - (0.7778892405807117 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.65d0) then
        tmp = ((x * 1.128386358070218d0) + 1d-9) + (x * (x * (-0.00011824294398844343d0)))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = ((x * 1.128386358070218) + 1e-9) + (x * (x * -0.00011824294398844343));
	} else {
		tmp = 1.0 - (0.7778892405807117 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = ((x * 1.128386358070218) + 1e-9) + (x * (x * -0.00011824294398844343))
	else:
		tmp = 1.0 - (0.7778892405807117 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(Float64(Float64(x * 1.128386358070218) + 1e-9) + Float64(x * Float64(x * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65)
		tmp = ((x * 1.128386358070218) + 1e-9) + (x * (x * -0.00011824294398844343));
	else
		tmp = 1.0 - (0.7778892405807117 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.65], N[(N[(N[(x * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision] + N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 72.4%

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

   3. Applied egg-rr 72.2%

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

      1. distribute-neg-frac 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. associate-+r+ 65.2%

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

      3. +-commutative 65.2%

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

      4. *-commutative 65.2%

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

      5. fma-def 65.2%

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

      6. *-commutative 65.2%

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

      7. unpow2 65.2%

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

      8. associate-*l* 65.2%

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

   8. Simplified 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)} \]
   9. Step-by-step derivation

      1. fma-udef 65.2%

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

   10. Applied egg-rr 65.2%

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

   if 1.6499999999999999 < 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. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 99.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\left(x \cdot 1.128386358070218 + 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \]

Alternative 7: 98.4% accurate, 121.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.65)
   (+ (* x 1.128386358070218) 1e-9)
   (- 1.0 (/ 0.7778892405807117 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = (x * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 - (0.7778892405807117 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.65d0) then
        tmp = (x * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = (x * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 - (0.7778892405807117 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = (x * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0 - (0.7778892405807117 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(Float64(x * 1.128386358070218) + 1e-9);
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65)
		tmp = (x * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0 - (0.7778892405807117 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.65], N[(N[(x * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 72.4%

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

   3. Applied egg-rr 72.2%

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

      1. distribute-neg-frac 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in x around 0 65.2%

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

      1. *-commutative 65.2%

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

   8. Simplified 65.2%

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

   if 1.6499999999999999 < 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. Applied egg-rr 100.0%

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

      1. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 99.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \]

Alternative 8: 53.0% accurate, 856.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 1e-9)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

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

Python

x = abs(x)
def code(x):
	return 1e-9

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1e-9

Derivation

1. Initial program 79.2%

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

3. Applied egg-rr 79.0%

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

   1. distribute-neg-frac 79.0%

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

5. Simplified 79.0%

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

6. Taylor expanded in x around 0 53.8%

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

7. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2023201 
(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)))))))