Jmat.Real.erf

Percentage Accurate: 79.2% → 86.5%

Time: 2.4min

Alternatives: 12

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% 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: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{t\_0}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}\\ t_2 := {t\_1}^{2}\\ \frac{\frac{{t\_1}^{4}}{1 + t\_2} + \frac{1}{-1 - t\_2}}{-1 + t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
              t_0))
            t_0))
          (* (exp (* x x)) (+ -1.0 (* (fabs x) -0.3275911)))))
        (t_2 (pow t_1 2.0)))
   (/ (+ (/ (pow t_1 4.0) (+ 1.0 t_2)) (/ 1.0 (- -1.0 t_2))) (+ -1.0 t_1))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * (-1.0 + (fabs(x) * -0.3275911)));
	double t_2 = pow(t_1, 2.0);
	return ((pow(t_1, 4.0) / (1.0 + t_2)) + (1.0 / (-1.0 - t_2))) / (-1.0 + t_1);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_1 = (0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_0)) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * ((-1.0d0) + (abs(x) * (-0.3275911d0))))
    t_2 = t_1 ** 2.0d0
    code = (((t_1 ** 4.0d0) / (1.0d0 + t_2)) + (1.0d0 / ((-1.0d0) - t_2))) / ((-1.0d0) + t_1)
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_1 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / (Math.exp((x * x)) * (-1.0 + (Math.abs(x) * -0.3275911)));
	double t_2 = Math.pow(t_1, 2.0);
	return ((Math.pow(t_1, 4.0) / (1.0 + t_2)) + (1.0 / (-1.0 - t_2))) / (-1.0 + t_1);
}

Python

def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	t_1 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / (math.exp((x * x)) * (-1.0 + (math.fabs(x) * -0.3275911)))
	t_2 = math.pow(t_1, 2.0)
	return ((math.pow(t_1, 4.0) / (1.0 + t_2)) + (1.0 / (-1.0 - t_2))) / (-1.0 + t_1)

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / Float64(exp(Float64(x * x)) * Float64(-1.0 + Float64(abs(x) * -0.3275911))))
	t_2 = t_1 ^ 2.0
	return Float64(Float64(Float64((t_1 ^ 4.0) / Float64(1.0 + t_2)) + Float64(1.0 / Float64(-1.0 - t_2))) / Float64(-1.0 + t_1))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * (-1.0 + (abs(x) * -0.3275911)));
	t_2 = t_1 ^ 2.0;
	tmp = (((t_1 ^ 4.0) / (1.0 + t_2)) + (1.0 / (-1.0 - t_2))) / (-1.0 + t_1);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

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

8. Applied egg-rr 86.6%

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

9. Final simplification 86.6%

   \[\leadsto \frac{\frac{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}\right)}^{4}}{1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}\right)}^{2}} + \frac{1}{-1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}\right)}^{2}}}{-1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}} \]
10. Add Preprocessing

Alternative 2: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := 0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{t\_0}\\ t_2 := \left|x\right| \cdot -0.3275911\\ t_3 := e^{x \cdot x} \cdot \left(-1 + t\_2\right)\\ t_4 := 1 - t\_2\\ \frac{1 + \frac{1}{\frac{{t\_3}^{3}}{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_4}}{t\_4}}{t\_4}}{t\_4}\right)}^{3}}}}{1 + \frac{-1 + \frac{t\_1}{t\_3}}{\frac{t\_3}{t\_1}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
             t_0))
           t_0)))
        (t_2 (* (fabs x) -0.3275911))
        (t_3 (* (exp (* x x)) (+ -1.0 t_2)))
        (t_4 (- 1.0 t_2)))
   (/
    (+
     1.0
     (/
      1.0
      (/
       (pow t_3 3.0)
       (pow
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_4)) t_4))
            t_4))
          t_4))
        3.0))))
    (+ 1.0 (/ (+ -1.0 (/ t_1 t_3)) (/ t_3 t_1))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0);
	double t_2 = fabs(x) * -0.3275911;
	double t_3 = exp((x * x)) * (-1.0 + t_2);
	double t_4 = 1.0 - t_2;
	return (1.0 + (1.0 / (pow(t_3, 3.0) / pow((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_4)) / t_4)) / t_4)) / t_4)), 3.0)))) / (1.0 + ((-1.0 + (t_1 / t_3)) / (t_3 / t_1)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_1 = 0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_0)) / t_0)) / t_0)) / t_0)
    t_2 = abs(x) * (-0.3275911d0)
    t_3 = exp((x * x)) * ((-1.0d0) + t_2)
    t_4 = 1.0d0 - t_2
    code = (1.0d0 + (1.0d0 / ((t_3 ** 3.0d0) / ((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_4)) / t_4)) / t_4)) / t_4)) ** 3.0d0)))) / (1.0d0 + (((-1.0d0) + (t_1 / t_3)) / (t_3 / t_1)))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0);
	double t_2 = Math.abs(x) * -0.3275911;
	double t_3 = Math.exp((x * x)) * (-1.0 + t_2);
	double t_4 = 1.0 - t_2;
	return (1.0 + (1.0 / (Math.pow(t_3, 3.0) / Math.pow((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_4)) / t_4)) / t_4)) / t_4)), 3.0)))) / (1.0 + ((-1.0 + (t_1 / t_3)) / (t_3 / t_1)));
}

Python

def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)
	t_2 = math.fabs(x) * -0.3275911
	t_3 = math.exp((x * x)) * (-1.0 + t_2)
	t_4 = 1.0 - t_2
	return (1.0 + (1.0 / (math.pow(t_3, 3.0) / math.pow((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_4)) / t_4)) / t_4)) / t_4)), 3.0)))) / (1.0 + ((-1.0 + (t_1 / t_3)) / (t_3 / t_1)))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / t_0))
	t_2 = Float64(abs(x) * -0.3275911)
	t_3 = Float64(exp(Float64(x * x)) * Float64(-1.0 + t_2))
	t_4 = Float64(1.0 - t_2)
	return Float64(Float64(1.0 + Float64(1.0 / Float64((t_3 ^ 3.0) / (Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_4)) / t_4)) / t_4)) / t_4)) ^ 3.0)))) / Float64(1.0 + Float64(Float64(-1.0 + Float64(t_1 / t_3)) / Float64(t_3 / t_1))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0);
	t_2 = abs(x) * -0.3275911;
	t_3 = exp((x * x)) * (-1.0 + t_2);
	t_4 = 1.0 - t_2;
	tmp = (1.0 + (1.0 / ((t_3 ^ 3.0) / ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_4)) / t_4)) / t_4)) / t_4)) ^ 3.0)))) / (1.0 + ((-1.0 + (t_1 / t_3)) / (t_3 / t_1)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - t$95$2), $MachinePrecision]}, N[(N[(1.0 + N[(1.0 / N[(N[Power[t$95$3, 3.0], $MachinePrecision] / N[Power[N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(-1.0 + N[(t$95$1 / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

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

8. Applied egg-rr 80.3%

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

9. Final simplification 80.3%

   \[\leadsto \frac{1 + \frac{1}{\frac{{\left(e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)\right)}^{3}}{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}\right)}^{3}}}}{1 + \frac{-1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}}{\frac{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}}} \]
10. Add Preprocessing

Alternative 3: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := 0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{t\_0}\\ t_2 := e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)\\ t_3 := \frac{t\_1}{t\_2}\\ \frac{1 + \frac{1}{\frac{1}{{t\_3}^{3}}}}{1 + \frac{-1 + t\_3}{\frac{t\_2}{t\_1}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
             t_0))
           t_0)))
        (t_2 (* (exp (* x x)) (+ -1.0 (* (fabs x) -0.3275911))))
        (t_3 (/ t_1 t_2)))
   (/
    (+ 1.0 (/ 1.0 (/ 1.0 (pow t_3 3.0))))
    (+ 1.0 (/ (+ -1.0 t_3) (/ t_2 t_1))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0);
	double t_2 = exp((x * x)) * (-1.0 + (fabs(x) * -0.3275911));
	double t_3 = t_1 / t_2;
	return (1.0 + (1.0 / (1.0 / pow(t_3, 3.0)))) / (1.0 + ((-1.0 + t_3) / (t_2 / t_1)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_1 = 0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_0)) / t_0)) / t_0)) / t_0)
    t_2 = exp((x * x)) * ((-1.0d0) + (abs(x) * (-0.3275911d0)))
    t_3 = t_1 / t_2
    code = (1.0d0 + (1.0d0 / (1.0d0 / (t_3 ** 3.0d0)))) / (1.0d0 + (((-1.0d0) + t_3) / (t_2 / t_1)))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0);
	double t_2 = Math.exp((x * x)) * (-1.0 + (Math.abs(x) * -0.3275911));
	double t_3 = t_1 / t_2;
	return (1.0 + (1.0 / (1.0 / Math.pow(t_3, 3.0)))) / (1.0 + ((-1.0 + t_3) / (t_2 / t_1)));
}

Python

def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)
	t_2 = math.exp((x * x)) * (-1.0 + (math.fabs(x) * -0.3275911))
	t_3 = t_1 / t_2
	return (1.0 + (1.0 / (1.0 / math.pow(t_3, 3.0)))) / (1.0 + ((-1.0 + t_3) / (t_2 / t_1)))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / t_0))
	t_2 = Float64(exp(Float64(x * x)) * Float64(-1.0 + Float64(abs(x) * -0.3275911)))
	t_3 = Float64(t_1 / t_2)
	return Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 / (t_3 ^ 3.0)))) / Float64(1.0 + Float64(Float64(-1.0 + t_3) / Float64(t_2 / t_1))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0);
	t_2 = exp((x * x)) * (-1.0 + (abs(x) * -0.3275911));
	t_3 = t_1 / t_2;
	tmp = (1.0 + (1.0 / (1.0 / (t_3 ^ 3.0)))) / (1.0 + ((-1.0 + t_3) / (t_2 / t_1)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, N[(N[(1.0 + N[(1.0 / N[(1.0 / N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(-1.0 + t$95$3), $MachinePrecision] / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

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

8. Applied egg-rr 80.3%

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

10. Applied egg-rr 80.2%

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

11. Final simplification 80.2%

    \[\leadsto \frac{1 + \frac{1}{\frac{1}{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}\right)}^{3}}}}{1 + \frac{-1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}}{\frac{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}}} \]
12. Add Preprocessing

Alternative 4: 79.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ t_2 := 1 - \left|x\right| \cdot -0.3275911\\ t_3 := e^{x \cdot x}\\ t_4 := t\_3 \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)\\ \frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{-1 - t\_0} - -1.453152027}{t\_1} - 1.421413741}{t\_1} - -0.284496736}{t\_1} - 0.254829592}{t\_4}\right)}^{2}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_1}}{t\_1}}{t\_1}}{t\_1}}{t\_4}} + \frac{t\_0}{\left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right) \cdot \frac{t\_3}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_2}}{t\_2}}{t\_2}}{t\_2}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) 0.3275911))
        (t_1 (+ 1.0 t_0))
        (t_2 (- 1.0 (* (fabs x) -0.3275911)))
        (t_3 (exp (* x x)))
        (t_4 (* t_3 (+ 1.0 (* x (* x -0.10731592879921))))))
   (+
    (/
     (-
      1.0
      (pow
       (/
        (-
         (/
          (-
           (/
            (-
             (/ (- (/ 1.061405429 (- -1.0 t_0)) -1.453152027) t_1)
             1.421413741)
            t_1)
           -0.284496736)
          t_1)
         0.254829592)
        t_4)
       2.0))
     (+
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1)) t_1))
         t_1))
       t_4)))
    (/
     t_0
     (*
      (+ 1.0 (* (* x x) -0.10731592879921))
      (/
       t_3
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_2)) t_2)) t_2))
         t_2))))))))

C

double code(double x) {
	double t_0 = fabs(x) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 - (fabs(x) * -0.3275911);
	double t_3 = exp((x * x));
	double t_4 = t_3 * (1.0 + (x * (x * -0.10731592879921)));
	return ((1.0 - pow((((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_4), 2.0)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_4))) + (t_0 / ((1.0 + ((x * x) * -0.10731592879921)) * (t_3 / (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_2)) / t_2)) / t_2)) / t_2)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = abs(x) * 0.3275911d0
    t_1 = 1.0d0 + t_0
    t_2 = 1.0d0 - (abs(x) * (-0.3275911d0))
    t_3 = exp((x * x))
    t_4 = t_3 * (1.0d0 + (x * (x * (-0.10731592879921d0))))
    code = ((1.0d0 - ((((((((((1.061405429d0 / ((-1.0d0) - t_0)) - (-1.453152027d0)) / t_1) - 1.421413741d0) / t_1) - (-0.284496736d0)) / t_1) - 0.254829592d0) / t_4) ** 2.0d0)) / (1.0d0 + ((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_1)) / t_1)) / t_1)) / t_1)) / t_4))) + (t_0 / ((1.0d0 + ((x * x) * (-0.10731592879921d0))) * (t_3 / (0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_2)) / t_2)) / t_2)) / t_2)))))
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 - (Math.abs(x) * -0.3275911);
	double t_3 = Math.exp((x * x));
	double t_4 = t_3 * (1.0 + (x * (x * -0.10731592879921)));
	return ((1.0 - Math.pow((((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_4), 2.0)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_4))) + (t_0 / ((1.0 + ((x * x) * -0.10731592879921)) * (t_3 / (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_2)) / t_2)) / t_2)) / t_2)))));
}

Python

def code(x):
	t_0 = math.fabs(x) * 0.3275911
	t_1 = 1.0 + t_0
	t_2 = 1.0 - (math.fabs(x) * -0.3275911)
	t_3 = math.exp((x * x))
	t_4 = t_3 * (1.0 + (x * (x * -0.10731592879921)))
	return ((1.0 - math.pow((((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_4), 2.0)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_4))) + (t_0 / ((1.0 + ((x * x) * -0.10731592879921)) * (t_3 / (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_2)) / t_2)) / t_2)) / t_2)))))

Julia

function code(x)
	t_0 = Float64(abs(x) * 0.3275911)
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(1.0 - Float64(abs(x) * -0.3275911))
	t_3 = exp(Float64(x * x))
	t_4 = Float64(t_3 * Float64(1.0 + Float64(x * Float64(x * -0.10731592879921))))
	return Float64(Float64(Float64(1.0 - (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / Float64(-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_4) ^ 2.0)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_4))) + Float64(t_0 / Float64(Float64(1.0 + Float64(Float64(x * x) * -0.10731592879921)) * Float64(t_3 / Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_2)) / t_2)) / t_2)) / t_2))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * 0.3275911;
	t_1 = 1.0 + t_0;
	t_2 = 1.0 - (abs(x) * -0.3275911);
	t_3 = exp((x * x));
	t_4 = t_3 * (1.0 + (x * (x * -0.10731592879921)));
	tmp = ((1.0 - ((((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_4) ^ 2.0)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_4))) + (t_0 / ((1.0 + ((x * x) * -0.10731592879921)) * (t_3 / (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_2)) / t_2)) / t_2)) / t_2)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(1.0 + N[(x * N[(x * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] - 0.254829592), $MachinePrecision] / t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \color{blue}{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)} \cdot \left(-1 + 0.3275911 \cdot \left|x\right|\right)} \]

8. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{\left(1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{e^{x \cdot x} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right)}\right)\right) + \frac{0.3275911 \cdot \left|x\right|}{\left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right) \cdot \frac{e^{x \cdot x}}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}}} \]

10. Applied egg-rr 79.1%

    \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-\left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)\right)}\right)}^{2}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)}}} + \frac{0.3275911 \cdot \left|x\right|}{\left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right) \cdot \frac{e^{x \cdot x}}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}} \]

11. Final simplification 79.1%

    \[\leadsto \frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{-1 - \left|x\right| \cdot 0.3275911} - -1.453152027}{1 + \left|x\right| \cdot 0.3275911} - 1.421413741}{1 + \left|x\right| \cdot 0.3275911} - -0.284496736}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)}\right)}^{2}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)}} + \frac{\left|x\right| \cdot 0.3275911}{\left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right) \cdot \frac{e^{x \cdot x}}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}} \]
12. Add Preprocessing

Alternative 5: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ t_2 := e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)\\ \frac{t\_0 \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_1}}{t\_1}}{t\_1}}{t\_1}\right)}{t\_2} + \left(1 + \frac{\frac{\frac{\frac{\frac{1.061405429}{-1 - t\_0} - -1.453152027}{t\_1} - 1.421413741}{t\_1} - -0.284496736}{t\_1} - 0.254829592}{t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) 0.3275911))
        (t_1 (+ 1.0 t_0))
        (t_2 (* (exp (* x x)) (+ 1.0 (* x (* x -0.10731592879921))))))
   (+
    (/
     (*
      t_0
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1)) t_1))
        t_1)))
     t_2)
    (+
     1.0
     (/
      (-
       (/
        (-
         (/
          (- (/ (- (/ 1.061405429 (- -1.0 t_0)) -1.453152027) t_1) 1.421413741)
          t_1)
         -0.284496736)
        t_1)
       0.254829592)
      t_2)))))

C

double code(double x) {
	double t_0 = fabs(x) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = exp((x * x)) * (1.0 + (x * (x * -0.10731592879921)));
	return ((t_0 * (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1))) / t_2) + (1.0 + (((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_2));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = abs(x) * 0.3275911d0
    t_1 = 1.0d0 + t_0
    t_2 = exp((x * x)) * (1.0d0 + (x * (x * (-0.10731592879921d0))))
    code = ((t_0 * (0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_1)) / t_1)) / t_1)) / t_1))) / t_2) + (1.0d0 + (((((((((1.061405429d0 / ((-1.0d0) - t_0)) - (-1.453152027d0)) / t_1) - 1.421413741d0) / t_1) - (-0.284496736d0)) / t_1) - 0.254829592d0) / t_2))
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = Math.exp((x * x)) * (1.0 + (x * (x * -0.10731592879921)));
	return ((t_0 * (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1))) / t_2) + (1.0 + (((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_2));
}

Python

def code(x):
	t_0 = math.fabs(x) * 0.3275911
	t_1 = 1.0 + t_0
	t_2 = math.exp((x * x)) * (1.0 + (x * (x * -0.10731592879921)))
	return ((t_0 * (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1))) / t_2) + (1.0 + (((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_2))

Julia

function code(x)
	t_0 = Float64(abs(x) * 0.3275911)
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(exp(Float64(x * x)) * Float64(1.0 + Float64(x * Float64(x * -0.10731592879921))))
	return Float64(Float64(Float64(t_0 * Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / t_1))) / t_2) + Float64(1.0 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / Float64(-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_2)))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * 0.3275911;
	t_1 = 1.0 + t_0;
	t_2 = exp((x * x)) * (1.0 + (x * (x * -0.10731592879921)));
	tmp = ((t_0 * (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1))) / t_2) + (1.0 + (((((((((1.061405429 / (-1.0 - t_0)) - -1.453152027) / t_1) - 1.421413741) / t_1) - -0.284496736) / t_1) - 0.254829592) / t_2));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(x * N[(x * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(1.0 + N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] - 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \color{blue}{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)} \cdot \left(-1 + 0.3275911 \cdot \left|x\right|\right)} \]

8. Applied egg-rr 79.1%

   \[\leadsto 1 + \color{blue}{\frac{\left(-1 - \left|x\right| \cdot -0.3275911\right) \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}\right)}{e^{x \cdot x} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right)}} \]

10. Applied egg-rr 79.1%

    \[\leadsto \color{blue}{\frac{\left(\left|x\right| \cdot 0.3275911\right) \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)} + \left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)}\right)} \]

11. Final simplification 79.1%

    \[\leadsto \frac{\left(\left|x\right| \cdot 0.3275911\right) \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)} + \left(1 + \frac{\frac{\frac{\frac{\frac{1.061405429}{-1 - \left|x\right| \cdot 0.3275911} - -1.453152027}{1 + \left|x\right| \cdot 0.3275911} - 1.421413741}{1 + \left|x\right| \cdot 0.3275911} - -0.284496736}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592}{e^{x \cdot x} \cdot \left(1 + x \cdot \left(x \cdot -0.10731592879921\right)\right)}\right) \]
12. Add Preprocessing

Alternative 6: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot -0.3275911\\ t_1 := 1 - t\_0\\ 1 + \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_1}}{t\_1}}{t\_1}}{t\_1}\right) \cdot \left(-1 - t\_0\right)}{e^{x \cdot x} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) -0.3275911)) (t_1 (- 1.0 t_0)))
   (+
    1.0
    (/
     (*
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1)) t_1))
        t_1))
      (- -1.0 t_0))
     (* (exp (* x x)) (+ 1.0 (* (* x x) -0.10731592879921)))))))

C

double code(double x) {
	double t_0 = fabs(x) * -0.3275911;
	double t_1 = 1.0 - t_0;
	return 1.0 + (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) * (-1.0 - t_0)) / (exp((x * x)) * (1.0 + ((x * x) * -0.10731592879921))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = abs(x) * (-0.3275911d0)
    t_1 = 1.0d0 - t_0
    code = 1.0d0 + (((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_1)) / t_1)) / t_1)) / t_1)) * ((-1.0d0) - t_0)) / (exp((x * x)) * (1.0d0 + ((x * x) * (-0.10731592879921d0)))))
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * -0.3275911;
	double t_1 = 1.0 - t_0;
	return 1.0 + (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) * (-1.0 - t_0)) / (Math.exp((x * x)) * (1.0 + ((x * x) * -0.10731592879921))));
}

Python

def code(x):
	t_0 = math.fabs(x) * -0.3275911
	t_1 = 1.0 - t_0
	return 1.0 + (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) * (-1.0 - t_0)) / (math.exp((x * x)) * (1.0 + ((x * x) * -0.10731592879921))))

Julia

function code(x)
	t_0 = Float64(abs(x) * -0.3275911)
	t_1 = Float64(1.0 - t_0)
	return Float64(1.0 + Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) * Float64(-1.0 - t_0)) / Float64(exp(Float64(x * x)) * Float64(1.0 + Float64(Float64(x * x) * -0.10731592879921)))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * -0.3275911;
	t_1 = 1.0 - t_0;
	tmp = 1.0 + (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) * (-1.0 - t_0)) / (exp((x * x)) * (1.0 + ((x * x) * -0.10731592879921))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, N[(1.0 + N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \color{blue}{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)} \cdot \left(-1 + 0.3275911 \cdot \left|x\right|\right)} \]

8. Applied egg-rr 79.1%

   \[\leadsto 1 + \color{blue}{\frac{\left(-1 - \left|x\right| \cdot -0.3275911\right) \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}\right)}{e^{x \cdot x} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right)}} \]

9. Final simplification 79.1%

   \[\leadsto 1 + \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}\right) \cdot \left(-1 - \left|x\right| \cdot -0.3275911\right)}{e^{x \cdot x} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.10731592879921\right)} \]
10. Add Preprocessing

Alternative 7: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ 1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429 + \left|x\right| \cdot -0.3477069720320819}{1 + x \cdot \left(x \cdot -0.10731592879921\right)}}{t\_0}}{t\_0}}{t\_0}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))))
   (+
    1.0
    (/
     (+
      0.254829592
      (/
       (+
        -0.284496736
        (/
         (+
          1.421413741
          (/
           (+
            -1.453152027
            (/
             (+ 1.061405429 (* (fabs x) -0.3477069720320819))
             (+ 1.0 (* x (* x -0.10731592879921)))))
           t_0))
         t_0))
       t_0))
     (* (exp (* x x)) (+ -1.0 (* (fabs x) -0.3275911)))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	return 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + ((1.061405429 + (fabs(x) * -0.3477069720320819)) / (1.0 + (x * (x * -0.10731592879921))))) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * (-1.0 + (fabs(x) * -0.3275911))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    code = 1.0d0 + ((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + ((1.061405429d0 + (abs(x) * (-0.3477069720320819d0))) / (1.0d0 + (x * (x * (-0.10731592879921d0)))))) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * ((-1.0d0) + (abs(x) * (-0.3275911d0)))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	return 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + ((1.061405429 + (Math.abs(x) * -0.3477069720320819)) / (1.0 + (x * (x * -0.10731592879921))))) / t_0)) / t_0)) / t_0)) / (Math.exp((x * x)) * (-1.0 + (Math.abs(x) * -0.3275911))));
}

Python

def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	return 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + ((1.061405429 + (math.fabs(x) * -0.3477069720320819)) / (1.0 + (x * (x * -0.10731592879921))))) / t_0)) / t_0)) / t_0)) / (math.exp((x * x)) * (-1.0 + (math.fabs(x) * -0.3275911))))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	return Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(Float64(1.061405429 + Float64(abs(x) * -0.3477069720320819)) / Float64(1.0 + Float64(x * Float64(x * -0.10731592879921))))) / t_0)) / t_0)) / t_0)) / Float64(exp(Float64(x * x)) * Float64(-1.0 + Float64(abs(x) * -0.3275911)))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	tmp = 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + ((1.061405429 + (abs(x) * -0.3477069720320819)) / (1.0 + (x * (x * -0.10731592879921))))) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * (-1.0 + (abs(x) * -0.3275911))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(N[(1.061405429 + N[(N[Abs[x], $MachinePrecision] * -0.3477069720320819), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(x * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right) + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}, 1 - \frac{3275911}{10000000} \cdot \left|x\right|, \frac{-1453152027}{1000000000}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}\right), \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right), \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 + \left|x\right| \cdot -0.3275911, -1.453152027\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} \]

8. Applied egg-rr 77.9%

   \[\leadsto 1 + \color{blue}{\frac{1}{\frac{-1 + \left|x\right| \cdot -0.3275911}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429 + \left(\left|x\right| \cdot -0.3275911\right) \cdot 1.061405429}{1 + \left(x \cdot x\right) \cdot -0.10731592879921}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{e^{x \cdot x}}}}} \]

10. Applied egg-rr 79.1%

    \[\leadsto \color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429 + \left|x\right| \cdot -0.3477069720320819}{1 + x \cdot \left(x \cdot -0.10731592879921\right)}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} + 1} \]

11. Final simplification 79.1%

    \[\leadsto 1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429 + \left|x\right| \cdot -0.3477069720320819}{1 + x \cdot \left(x \cdot -0.10731592879921\right)}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} \]
12. Add Preprocessing

Alternative 8: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot -0.3275911\\ t_1 := 1 - t\_0\\ \frac{1}{\frac{1}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_1}}{t\_1}}{t\_1}}{t\_1}}{e^{x \cdot x} \cdot \left(-1 + t\_0\right)}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) -0.3275911)) (t_1 (- 1.0 t_0)))
   (/
    1.0
    (/
     1.0
     (+
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1)) t_1))
         t_1))
       (* (exp (* x x)) (+ -1.0 t_0))))))))

C

double code(double x) {
	double t_0 = fabs(x) * -0.3275911;
	double t_1 = 1.0 - t_0;
	return 1.0 / (1.0 / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / (exp((x * x)) * (-1.0 + t_0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = abs(x) * (-0.3275911d0)
    t_1 = 1.0d0 - t_0
    code = 1.0d0 / (1.0d0 / (1.0d0 + ((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((-1.453152027d0) + (1.061405429d0 / t_1)) / t_1)) / t_1)) / t_1)) / (exp((x * x)) * ((-1.0d0) + t_0)))))
end function

Java

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

Python

def code(x):
	t_0 = math.fabs(x) * -0.3275911
	t_1 = 1.0 - t_0
	return 1.0 / (1.0 / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / (math.exp((x * x)) * (-1.0 + t_0)))))

Julia

function code(x)
	t_0 = Float64(abs(x) * -0.3275911)
	t_1 = Float64(1.0 - t_0)
	return Float64(1.0 / Float64(1.0 / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / Float64(exp(Float64(x * x)) * Float64(-1.0 + t_0))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * -0.3275911;
	t_1 = 1.0 - t_0;
	tmp = 1.0 / (1.0 / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / (exp((x * x)) * (-1.0 + t_0)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, N[(1.0 / N[(1.0 / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

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

8. Applied egg-rr 79.1%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

5. Final simplification 79.1%

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

Alternative 10: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot -0.3275911\\ t_1 := 1 - t\_0\\ 1 + \frac{1}{\frac{-1 + t\_0}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\left|x\right| \cdot -0.3477069720320819 + -0.391746598}{t\_1}}{t\_1}}{t\_1}}{e^{x \cdot x}}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) -0.3275911)) (t_1 (- 1.0 t_0)))
   (+
    1.0
    (/
     1.0
     (/
      (+ -1.0 t_0)
      (/
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/ (+ (* (fabs x) -0.3477069720320819) -0.391746598) t_1))
           t_1))
         t_1))
       (exp (* x x))))))))

C

double code(double x) {
	double t_0 = fabs(x) * -0.3275911;
	double t_1 = 1.0 - t_0;
	return 1.0 + (1.0 / ((-1.0 + t_0) / ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((fabs(x) * -0.3477069720320819) + -0.391746598) / t_1)) / t_1)) / t_1)) / exp((x * x)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = abs(x) * (-0.3275911d0)
    t_1 = 1.0d0 - t_0
    code = 1.0d0 + (1.0d0 / (((-1.0d0) + t_0) / ((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((abs(x) * (-0.3477069720320819d0)) + (-0.391746598d0)) / t_1)) / t_1)) / t_1)) / exp((x * x)))))
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * -0.3275911;
	double t_1 = 1.0 - t_0;
	return 1.0 + (1.0 / ((-1.0 + t_0) / ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((Math.abs(x) * -0.3477069720320819) + -0.391746598) / t_1)) / t_1)) / t_1)) / Math.exp((x * x)))));
}

Python

def code(x):
	t_0 = math.fabs(x) * -0.3275911
	t_1 = 1.0 - t_0
	return 1.0 + (1.0 / ((-1.0 + t_0) / ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((math.fabs(x) * -0.3477069720320819) + -0.391746598) / t_1)) / t_1)) / t_1)) / math.exp((x * x)))))

Julia

function code(x)
	t_0 = Float64(abs(x) * -0.3275911)
	t_1 = Float64(1.0 - t_0)
	return Float64(1.0 + Float64(1.0 / Float64(Float64(-1.0 + t_0) / Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(Float64(abs(x) * -0.3477069720320819) + -0.391746598) / t_1)) / t_1)) / t_1)) / exp(Float64(x * x))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * -0.3275911;
	t_1 = 1.0 - t_0;
	tmp = 1.0 + (1.0 / ((-1.0 + t_0) / ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((abs(x) * -0.3477069720320819) + -0.391746598) / t_1)) / t_1)) / t_1)) / exp((x * x)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, N[(1.0 + N[(1.0 / N[(N[(-1.0 + t$95$0), $MachinePrecision] / N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(N[(N[Abs[x], $MachinePrecision] * -0.3477069720320819), $MachinePrecision] + -0.391746598), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right) + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}, 1 - \frac{3275911}{10000000} \cdot \left|x\right|, \frac{-1453152027}{1000000000}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}\right), \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right), \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 + \left|x\right| \cdot -0.3275911, -1.453152027\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} \]

8. Applied egg-rr 77.9%

   \[\leadsto 1 + \color{blue}{\frac{1}{\frac{-1 + \left|x\right| \cdot -0.3275911}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429 + \left(\left|x\right| \cdot -0.3275911\right) \cdot 1.061405429}{1 + \left(x \cdot x\right) \cdot -0.10731592879921}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{e^{x \cdot x}}}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\color{blue}{\left(\frac{-3477069720320819}{10000000000000000} \cdot \left|x\right| - \frac{195873299}{500000000}\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{-3477069720320819}{10000000000000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-3477069720320819}{10000000000000000} \cdot \left|x\right|\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right| \cdot \frac{-3477069720320819}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left|x\right|\right), \frac{-3477069720320819}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    5. fabs-lowering-fabs.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3477069720320819}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    6. metadata-eval 77.3%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3477069720320819}{10000000000000000}\right), \frac{-195873299}{500000000}\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

11. Simplified 77.3%

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

Alternative 11: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ 1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\left|x\right| \cdot -0.3477069720320819 + -0.391746598}{t\_0}}{t\_0}}{t\_0}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))))
   (+
    1.0
    (/
     (+
      0.254829592
      (/
       (+
        -0.284496736
        (/
         (+
          1.421413741
          (/ (+ (* (fabs x) -0.3477069720320819) -0.391746598) t_0))
         t_0))
       t_0))
     (* (exp (* x x)) (+ -1.0 (* (fabs x) -0.3275911)))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	return 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((fabs(x) * -0.3477069720320819) + -0.391746598) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * (-1.0 + (fabs(x) * -0.3275911))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    code = 1.0d0 + ((0.254829592d0 + (((-0.284496736d0) + ((1.421413741d0 + (((abs(x) * (-0.3477069720320819d0)) + (-0.391746598d0)) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * ((-1.0d0) + (abs(x) * (-0.3275911d0)))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	return 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((Math.abs(x) * -0.3477069720320819) + -0.391746598) / t_0)) / t_0)) / t_0)) / (Math.exp((x * x)) * (-1.0 + (Math.abs(x) * -0.3275911))));
}

Python

def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	return 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((math.fabs(x) * -0.3477069720320819) + -0.391746598) / t_0)) / t_0)) / t_0)) / (math.exp((x * x)) * (-1.0 + (math.fabs(x) * -0.3275911))))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	return Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(Float64(abs(x) * -0.3477069720320819) + -0.391746598) / t_0)) / t_0)) / t_0)) / Float64(exp(Float64(x * x)) * Float64(-1.0 + Float64(abs(x) * -0.3275911)))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	tmp = 1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + (((abs(x) * -0.3477069720320819) + -0.391746598) / t_0)) / t_0)) / t_0)) / (exp((x * x)) * (-1.0 + (abs(x) * -0.3275911))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(N[(N[Abs[x], $MachinePrecision] * -0.3477069720320819), $MachinePrecision] + -0.391746598), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + N[(N[Abs[x], $MachinePrecision] * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right) + \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}, 1 - \frac{3275911}{10000000} \cdot \left|x\right|, \frac{-1453152027}{1000000000}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}\right), \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right), \frac{-1453152027}{1000000000}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{3275911}{10000000}, \mathsf{fabs.f64}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 + \left|x\right| \cdot -0.3275911, -1.453152027\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x} \cdot \left(-1 + \left|x\right| \cdot -0.3275911\right)} \]

8. Applied egg-rr 77.9%

   \[\leadsto 1 + \color{blue}{\frac{1}{\frac{-1 + \left|x\right| \cdot -0.3275911}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429 + \left(\left|x\right| \cdot -0.3275911\right) \cdot 1.061405429}{1 + \left(x \cdot x\right) \cdot -0.10731592879921}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{1 - \left|x\right| \cdot -0.3275911}}{e^{x \cdot x}}}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\color{blue}{\left(\frac{-3477069720320819}{10000000000000000} \cdot \left|x\right| - \frac{195873299}{500000000}\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\left(\frac{-3477069720320819}{10000000000000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-3477069720320819}{10000000000000000} \cdot \left|x\right|\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right| \cdot \frac{-3477069720320819}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left|x\right|\right), \frac{-3477069720320819}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    5. fabs-lowering-fabs.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3477069720320819}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{195873299}{500000000}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    6. metadata-eval 77.3%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{31853699}{125000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-8890523}{31250000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1421413741}{1000000000}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3477069720320819}{10000000000000000}\right), \frac{-195873299}{500000000}\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(x\right), \frac{-3275911}{10000000}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

11. Simplified 77.3%

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

13. Applied egg-rr 76.6%

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

14. Final simplification 76.6%

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

Alternative 12: 55.7% accurate, 856.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

6. Applied egg-rr 79.1%

   \[\leadsto 1 + \color{blue}{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)} \cdot \left(-1 + 0.3275911 \cdot \left|x\right|\right)} \]

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
8. Step-by-step derivation

9. Simplified 55.9%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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)))))))