Jmat.Real.erf

Percentage Accurate: 79.3% → 99.6%

Time: 19.1s

Alternatives: 8

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_0}}{t_0}}{t_0}}{e^{x \cdot x}}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 1e-9)
     (/
      (- 1e-18 (* (pow x 2.0) 1.2732557730789702))
      (+ 1e-9 (* x -1.128386358070218)))
     (-
      1.0
      (/
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/ (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0))) t_0))
            t_0))
          t_0))
        (exp (* x x)))
       t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 1e-9) {
		tmp = (1e-18 - (pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / t_0)) / exp((x * x))) / t_0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 1e-9)
		tmp = Float64(Float64(1e-18 - Float64((x ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x * -1.128386358070218)));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / t_0)) / exp(Float64(x * x))) / t_0));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-9], N[(N[(1e-18 - N[(N[Power[x, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 93.1%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 93.1%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 95.8%

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

      2. rem-cbrt-cube 97.3%

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

      3. flip-+ 97.3%

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

      4. metadata-eval 97.3%

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

      5. pow2 97.3%

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

   10. Applied egg-rr 97.3%

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

       1. unpow2 97.3%

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

       2. swap-sqr 97.3%

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

       3. unpow2 97.3%

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

       4. metadata-eval 97.3%

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

       5. sub-neg 97.3%

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

       6. distribute-rgt-neg-in 97.3%

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

       7. metadata-eval 97.3%

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

   12. Simplified 97.3%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. unpow1 99.7%

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

      4. sqr-pow 55.5%

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

      5. fabs-sqr 55.5%

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

      6. sqr-pow 99.3%

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

      7. unpow1 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{x \cdot x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(\sqrt[3]{{\left(1.421413741 + \frac{\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}} \cdot \frac{-1}{t_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 1e-9)
     (/
      (- 1e-18 (* (pow x 2.0) 1.2732557730789702))
      (+ 1e-9 (* x -1.128386358070218)))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            (cbrt
             (pow
              (+
               1.421413741
               (/
                (fma 1.061405429 (/ 1.0 (fma 0.3275911 x 1.0)) -1.453152027)
                (fma 0.3275911 x 1.0)))
              3.0))
            (/ -1.0 t_0))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 1e-9) {
		tmp = (1e-18 - (pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_1 * ((t_1 * ((cbrt(pow((1.421413741 + (fma(1.061405429, (1.0 / fma(0.3275911, x, 1.0)), -1.453152027) / fma(0.3275911, x, 1.0))), 3.0)) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 1e-9)
		tmp = Float64(Float64(1e-18 - Float64((x ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x * -1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(cbrt((Float64(1.421413741 + Float64(fma(1.061405429, Float64(1.0 / fma(0.3275911, x, 1.0)), -1.453152027) / fma(0.3275911, x, 1.0))) ^ 3.0)) * Float64(-1.0 / t_0)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-9], N[(N[(1e-18 - N[(N[Power[x, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(N[Power[N[Power[N[(1.421413741 + N[(N[(1.061405429 * N[(1.0 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 93.1%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 93.1%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 95.8%

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

      2. rem-cbrt-cube 97.3%

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

      3. flip-+ 97.3%

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

      4. metadata-eval 97.3%

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

      5. pow2 97.3%

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

   10. Applied egg-rr 97.3%

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

       1. unpow2 97.3%

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

       2. swap-sqr 97.3%

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

       3. unpow2 97.3%

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

       4. metadata-eval 97.3%

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

       5. sub-neg 97.3%

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

       6. distribute-rgt-neg-in 97.3%

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

       7. metadata-eval 97.3%

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

   12. Simplified 97.3%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-udef 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-udef 99.7%

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

      7. add-cbrt-cube 99.7%

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

      8. pow3 99.7%

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

   5. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. div-inv 99.2%

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

      3. fma-def 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\sqrt[3]{{\left(1.421413741 + \frac{\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}} \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 1e-9)
     (/
      (- 1e-18 (* (pow x 2.0) 1.2732557730789702))
      (+ 1e-9 (* x -1.128386358070218)))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_1
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              t_1
              (fma 1.061405429 (/ 1.0 (fma 0.3275911 x 1.0)) -1.453152027)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 1e-9) {
		tmp = (1e-18 - (pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_1 * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * fma(1.061405429, (1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 1e-9)
		tmp = Float64(Float64(1e-18 - Float64((x ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x * -1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_1 * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * fma(1.061405429, Float64(1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-9], N[(N[(1e-18 - N[(N[Power[x, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(1.061405429 * N[(1.0 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 93.1%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 93.1%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 95.8%

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

      2. rem-cbrt-cube 97.3%

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

      3. flip-+ 97.3%

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

      4. metadata-eval 97.3%

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

      5. pow2 97.3%

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

   10. Applied egg-rr 97.3%

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

       1. unpow2 97.3%

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

       2. swap-sqr 97.3%

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

       3. unpow2 97.3%

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

       4. metadata-eval 97.3%

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

       5. sub-neg 97.3%

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

       6. distribute-rgt-neg-in 97.3%

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

       7. metadata-eval 97.3%

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

   12. Simplified 97.3%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.7%

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

      3. fma-def 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-udef 99.7%

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

      6. add-sqr-sqrt 55.4%

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

      7. fabs-sqr 55.4%

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

      8. add-sqr-sqrt 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_2 \cdot \left(t_2 \cdot \left(\left(1.421413741 + \frac{1}{t_0} \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right) \cdot \frac{-1}{t_1} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= x 1.1e-6)
     (/
      (- 1e-18 (* (pow x 2.0) 1.2732557730789702))
      (+ 1e-9 (* x -1.128386358070218)))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_2
        (-
         (*
          t_2
          (-
           (*
            (+
             1.421413741
             (* (/ 1.0 t_0) (+ -1.453152027 (/ 1.061405429 t_0))))
            (/ -1.0 t_1))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 + (fabs(x) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (x <= 1.1e-6) {
		tmp = (1e-18 - (pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_2 * ((t_2 * (((1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0)))) * (-1.0 / t_1)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    t_1 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_2 = 1.0d0 / t_1
    if (x <= 1.1d-6) then
        tmp = (1d-18 - ((x ** 2.0d0) * 1.2732557730789702d0)) / (1d-9 + (x * (-1.128386358070218d0)))
    else
        tmp = 1.0d0 + (exp((x * -x)) * (t_2 * ((t_2 * (((1.421413741d0 + ((1.0d0 / t_0) * ((-1.453152027d0) + (1.061405429d0 / t_0)))) * ((-1.0d0) / t_1)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (x <= 1.1e-6) {
		tmp = (1e-18 - (Math.pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = 1.0 + (Math.exp((x * -x)) * (t_2 * ((t_2 * (((1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0)))) * (-1.0 / t_1)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	t_1 = 1.0 + (math.fabs(x) * 0.3275911)
	t_2 = 1.0 / t_1
	tmp = 0
	if x <= 1.1e-6:
		tmp = (1e-18 - (math.pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218))
	else:
		tmp = 1.0 + (math.exp((x * -x)) * (t_2 * ((t_2 * (((1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0)))) * (-1.0 / t_1)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (x <= 1.1e-6)
		tmp = Float64(Float64(1e-18 - Float64((x ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x * -1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_2 * Float64(Float64(t_2 * Float64(Float64(Float64(1.421413741 + Float64(Float64(1.0 / t_0) * Float64(-1.453152027 + Float64(1.061405429 / t_0)))) * Float64(-1.0 / t_1)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	t_1 = 1.0 + (abs(x) * 0.3275911);
	t_2 = 1.0 / t_1;
	tmp = 0.0;
	if (x <= 1.1e-6)
		tmp = (1e-18 - ((x ^ 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	else
		tmp = 1.0 + (exp((x * -x)) * (t_2 * ((t_2 * (((1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0)))) * (-1.0 / t_1)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[x, 1.1e-6], N[(N[(1e-18 - N[(N[Power[x, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[(N[(t$95$2 * N[(N[(N[(1.421413741 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-6

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 65.7%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 65.7%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 65.7%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 67.2%

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

      2. rem-cbrt-cube 68.3%

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

      3. flip-+ 68.3%

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

      4. metadata-eval 68.3%

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

      5. pow2 68.3%

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

   10. Applied egg-rr 68.3%

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

       1. unpow2 68.3%

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

       2. swap-sqr 68.3%

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

       3. unpow2 68.3%

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

       4. metadata-eval 68.3%

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

       5. sub-neg 68.3%

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

       6. distribute-rgt-neg-in 68.3%

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

       7. metadata-eval 68.3%

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

   12. Simplified 68.3%

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

   if 1.1000000000000001e-6 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 5: 54.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 900000000:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{10^{-27}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 900000000.0)
   (/
    (- 1e-18 (* (pow x 2.0) 1.2732557730789702))
    (+ 1e-9 (* x -1.128386358070218)))
   (cbrt 1e-27)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 900000000.0) {
		tmp = (1e-18 - (pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = cbrt(1e-27);
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 900000000.0) {
		tmp = (1e-18 - (Math.pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = Math.cbrt(1e-27);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 900000000.0)
		tmp = Float64(Float64(1e-18 - Float64((x ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x * -1.128386358070218)));
	else
		tmp = cbrt(1e-27);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 900000000.0], N[(N[(1e-18 - N[(N[Power[x, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[1e-27, 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e8

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 65.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 65.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 67.0%

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

      2. rem-cbrt-cube 68.0%

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

      3. flip-+ 68.0%

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

      4. metadata-eval 68.0%

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

      5. pow2 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. unpow2 68.0%

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

       2. swap-sqr 68.0%

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

       3. unpow2 68.0%

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

       4. metadata-eval 68.0%

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

       5. sub-neg 68.0%

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

       6. distribute-rgt-neg-in 68.0%

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

       7. metadata-eval 68.0%

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

   12. Simplified 68.0%

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

   if 9e8 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 11.1%

      \[\leadsto \color{blue}{\sqrt[3]{10^{-27}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 900000000:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{10^{-27}}\\ \end{array} \]

Alternative 6: 99.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89)
   (/
    (- 1e-18 (* (pow x 2.0) 1.2732557730789702))
    (+ 1e-9 (* x -1.128386358070218)))
   (pow 1.0 0.3333333333333333)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = (1e-18 - (pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = pow(1.0, 0.3333333333333333);
	}
	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 <= 0.89d0) then
        tmp = (1d-18 - ((x ** 2.0d0) * 1.2732557730789702d0)) / (1d-9 + (x * (-1.128386358070218d0)))
    else
        tmp = 1.0d0 ** 0.3333333333333333d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = (1e-18 - (Math.pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	} else {
		tmp = Math.pow(1.0, 0.3333333333333333);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.89:
		tmp = (1e-18 - (math.pow(x, 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218))
	else:
		tmp = math.pow(1.0, 0.3333333333333333)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(Float64(1e-18 - Float64((x ^ 2.0) * 1.2732557730789702)) / Float64(1e-9 + Float64(x * -1.128386358070218)));
	else
		tmp = 1.0 ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = (1e-18 - ((x ^ 2.0) * 1.2732557730789702)) / (1e-9 + (x * -1.128386358070218));
	else
		tmp = 1.0 ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(N[(1e-18 - N[(N[Power[x, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(1e-9 + N[(x * -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[1.0, 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 65.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 65.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 67.0%

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

      2. rem-cbrt-cube 68.0%

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

      3. flip-+ 68.0%

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

      4. metadata-eval 68.0%

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

      5. pow2 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. unpow2 68.0%

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

       2. swap-sqr 68.0%

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

       3. unpow2 68.0%

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

       4. metadata-eval 68.0%

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

       5. sub-neg 68.0%

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

       6. distribute-rgt-neg-in 68.0%

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

       7. metadata-eval 68.0%

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

   12. Simplified 68.0%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto {\left({\color{blue}{1}}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;\frac{10^{-18} - {x}^{2} \cdot 1.2732557730789702}{10^{-9} + x \cdot -1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 7: 54.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 900000000:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{10^{-27}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 900000000.0) (+ 1e-9 (* x 1.128386358070218)) (cbrt 1e-27)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 900000000.0) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = cbrt(1e-27);
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 900000000.0) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = Math.cbrt(1e-27);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 900000000.0)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = cbrt(1e-27);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 900000000.0], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[Power[1e-27, 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e8

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 65.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 65.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
   9. Step-by-step derivation

      1. unpow1/3 67.0%

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

      2. rem-cbrt-cube 68.0%

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

      3. +-commutative 68.0%

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

   10. Applied egg-rr 68.0%

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

   if 9e8 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 11.1%

      \[\leadsto \color{blue}{\sqrt[3]{10^{-27}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 900000000:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{10^{-27}}\\ \end{array} \]

Alternative 8: 51.8% accurate, 171.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 10^{-9} + x \cdot 1.128386358070218 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (+ 1e-9 (* x 1.128386358070218)))

C

x = abs(x);
double code(double x) {
	return 1e-9 + (x * 1.128386358070218);
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9 + (x * 1.128386358070218d0)
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 1e-9 + (x * 1.128386358070218);
}

Python

x = abs(x)
def code(x):
	return 1e-9 + (x * 1.128386358070218)

Julia

x = abs(x)
function code(x)
	return Float64(1e-9 + Float64(x * 1.128386358070218))
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 1e-9 + (x * 1.128386358070218);
end

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

5. Applied egg-rr 30.6%

   \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot e^{{x}^{2}}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in x around 0 49.1%

   \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]
7. Step-by-step derivation

   1. *-commutative 49.1%

      \[\leadsto {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

8. Simplified 49.1%

   \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
9. Step-by-step derivation

   1. unpow1/3 50.2%

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

   2. rem-cbrt-cube 51.2%

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

   3. +-commutative 51.2%

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

10. Applied egg-rr 51.2%

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

11. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2023310 
(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)))))))