Jmat.Real.erf

Percentage Accurate: 79.1% → 98.5%

Time: 49.5s

Alternatives: 7

Speedup: 3.4×

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: 79.1% 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: 98.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{t\_0}}{e^{x \cdot x} \cdot \left(-1 - x \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911))))
   (if (<= (fabs x) 1e-11)
     (+ 1e-9 (* x 1.128386358070218))
     (+
      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 (* x 0.3275911))))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double tmp;
	if (fabs(x) <= 1e-11) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		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 - (x * 0.3275911))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    if (abs(x) <= 1d-11) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 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) - (x * 0.3275911d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double tmp;
	if (Math.abs(x) <= 1e-11) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 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 - (x * 0.3275911))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	tmp = 0
	if math.fabs(x) <= 1e-11:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 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 - (x * 0.3275911))))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 1e-11)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 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(x * 0.3275911)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	tmp = 0.0;
	if (abs(x) <= 1e-11)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		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 - (x * 0.3275911))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-11], N[(1e-9 + N[(x * 1.128386358070218), $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[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.99999999999999939e-12

   1. Initial program 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 9.99999999999999939e-12 < (fabs.f64 x)

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{e^{x \cdot x} \cdot \left(-1 - x \cdot 0.3275911\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\frac{1.029667143}{-1 - x \cdot 0.3275911} - -0.284496736}{t\_0} - 0.254829592}{t\_0 \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911))))
   (if (<= x 0.5)
     (+ 1e-9 (* x 1.128386358070218))
     (+
      1.0
      (/
       (-
        (/ (- (/ 1.029667143 (- -1.0 (* x 0.3275911))) -0.284496736) t_0)
        0.254829592)
       (* t_0 (exp (* x x))))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.5) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (((((1.029667143 / (-1.0 - (x * 0.3275911))) - -0.284496736) / t_0) - 0.254829592) / (t_0 * exp((x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    if (x <= 0.5d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (((((1.029667143d0 / ((-1.0d0) - (x * 0.3275911d0))) - (-0.284496736d0)) / t_0) - 0.254829592d0) / (t_0 * exp((x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.5) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (((((1.029667143 / (-1.0 - (x * 0.3275911))) - -0.284496736) / t_0) - 0.254829592) / (t_0 * Math.exp((x * x))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	tmp = 0
	if x <= 0.5:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 + (((((1.029667143 / (-1.0 - (x * 0.3275911))) - -0.284496736) / t_0) - 0.254829592) / (t_0 * math.exp((x * x))))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(1.029667143 / Float64(-1.0 - Float64(x * 0.3275911))) - -0.284496736) / t_0) - 0.254829592) / Float64(t_0 * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	tmp = 0.0;
	if (x <= 0.5)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 + (((((1.029667143 / (-1.0 - (x * 0.3275911))) - -0.284496736) / t_0) - 0.254829592) / (t_0 * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.5], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(1.029667143 / N[(-1.0 - N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   7. Simplified 67.9%

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

   if 0.5 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 99.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\frac{1.029667143}{-1 - x \cdot 0.3275911} - -0.284496736}{1 + x \cdot 0.3275911} - 0.254829592}{\left(1 + x \cdot 0.3275911\right) \cdot e^{x \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x \cdot -0.3373097920092273 + 0.745170407}{-1 - x \cdot 0.3275911} - 0.254829592}{\left(1 + x \cdot 0.3275911\right) \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.42)
   (+ 1e-9 (* x 1.128386358070218))
   (+
    1.0
    (/
     (-
      (/ (+ (* x -0.3373097920092273) 0.745170407) (- -1.0 (* x 0.3275911)))
      0.254829592)
     (* (+ 1.0 (* x 0.3275911)) (exp (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.42) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (((((x * -0.3373097920092273) + 0.745170407) / (-1.0 - (x * 0.3275911))) - 0.254829592) / ((1.0 + (x * 0.3275911)) * exp((x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.42d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (((((x * (-0.3373097920092273d0)) + 0.745170407d0) / ((-1.0d0) - (x * 0.3275911d0))) - 0.254829592d0) / ((1.0d0 + (x * 0.3275911d0)) * exp((x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.42) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (((((x * -0.3373097920092273) + 0.745170407) / (-1.0 - (x * 0.3275911))) - 0.254829592) / ((1.0 + (x * 0.3275911)) * Math.exp((x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.42:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 + (((((x * -0.3373097920092273) + 0.745170407) / (-1.0 - (x * 0.3275911))) - 0.254829592) / ((1.0 + (x * 0.3275911)) * math.exp((x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.42)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(x * -0.3373097920092273) + 0.745170407) / Float64(-1.0 - Float64(x * 0.3275911))) - 0.254829592) / Float64(Float64(1.0 + Float64(x * 0.3275911)) * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.42)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 + (((((x * -0.3373097920092273) + 0.745170407) / (-1.0 - (x * 0.3275911))) - 0.254829592) / ((1.0 + (x * 0.3275911)) * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.42], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(x * -0.3373097920092273), $MachinePrecision] + 0.745170407), $MachinePrecision] / N[(-1.0 - N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[(N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.419999999999999984

   1. Initial program 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   7. Simplified 67.9%

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

   if 0.419999999999999984 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143}}{1 + 0.3275911 \cdot x}}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

   6. Taylor expanded in x around 0 99.1%

      \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \color{blue}{\left(1.029667143 + -0.3373097920092273 \cdot x\right)}}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \left(1.029667143 + \color{blue}{x \cdot -0.3373097920092273}\right)}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

   8. Simplified 99.1%

      \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \color{blue}{\left(1.029667143 + x \cdot -0.3373097920092273\right)}}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 99.1%

         \[\leadsto 1 - \frac{\color{blue}{1 \cdot \left(0.254829592 + \frac{-0.284496736 + \left(1.029667143 + x \cdot -0.3373097920092273\right)}{1 + 0.3275911 \cdot x}\right)}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

      2. *-un-lft-identity 99.1%

         \[\leadsto 1 - \frac{\color{blue}{0.254829592 + \frac{-0.284496736 + \left(1.029667143 + x \cdot -0.3373097920092273\right)}{1 + 0.3275911 \cdot x}}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

      3. associate-+r+ 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \frac{\color{blue}{\left(-0.284496736 + 1.029667143\right) + x \cdot -0.3373097920092273}}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

      4. +-commutative 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \frac{\color{blue}{x \cdot -0.3373097920092273 + \left(-0.284496736 + 1.029667143\right)}}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

      5. metadata-eval 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \frac{x \cdot -0.3373097920092273 + \color{blue}{0.745170407}}{1 + 0.3275911 \cdot x}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

      6. *-commutative 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \frac{x \cdot -0.3373097920092273 + 0.745170407}{1 + \color{blue}{x \cdot 0.3275911}}}{\left(1 + 0.3275911 \cdot x\right) \cdot e^{x \cdot x}} \]

      7. *-commutative 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \frac{x \cdot -0.3373097920092273 + 0.745170407}{1 + x \cdot 0.3275911}}{\left(1 + \color{blue}{x \cdot 0.3275911}\right) \cdot e^{x \cdot x}} \]

   10. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{x \cdot -0.3373097920092273 + 0.745170407}{1 + x \cdot 0.3275911}}{\left(1 + x \cdot 0.3275911\right) \cdot e^{x \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x \cdot -0.3373097920092273 + 0.745170407}{-1 - x \cdot 0.3275911} - 0.254829592}{\left(1 + x \cdot 0.3275911\right) \cdot e^{x \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (+ 1e-9 (* x 1.128386358070218))
   (+ 1.0 (/ -0.7778892405807117 (* x (exp (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (-0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 + ((-0.7778892405807117d0) / (x * exp((x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (-0.7778892405807117 / (x * Math.exp((x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.88:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 + (-0.7778892405807117 / (x * math.exp((x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(-0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 + (-0.7778892405807117 / (x * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.88], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   7. Simplified 67.9%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. sub-neg 99.0%

         \[\leadsto \color{blue}{1 + \left(-0.7778892405807117 \cdot \frac{1}{x \cdot e^{{x}^{2}}}\right)} \]

      2. associate-*r/ 99.0%

         \[\leadsto 1 + \left(-\color{blue}{\frac{0.7778892405807117 \cdot 1}{x \cdot e^{{x}^{2}}}}\right) \]

      3. metadata-eval 99.0%

         \[\leadsto 1 + \left(-\frac{\color{blue}{0.7778892405807117}}{x \cdot e^{{x}^{2}}}\right) \]

      4. distribute-neg-frac 99.0%

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

      5. metadata-eval 99.0%

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

      6. unpow2 99.0%

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

   7. Simplified 99.0%

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

Alternative 5: 73.5% accurate, 85.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 0.88], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   7. Simplified 67.9%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

Alternative 6: 75.2% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0 70.5%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

Alternative 7: 53.3% accurate, 856.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1e-9)

C

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

Fortran

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

Java

public static double code(double x) {
	return 1e-9;
}

Python

def code(x):
	return 1e-9

Julia

function code(x)
	return 1e-9
end

MATLAB

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

Wolfram

code[x_] := 1e-9

Derivation

1. Initial program 79.4%

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in x around 0 54.0%

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

Reproduce

herbie shell --seed 2024107 
(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)))))))