Jmat.Real.erf

Percentage Accurate: 78.9% → 99.1%

Time: 21.3s

Alternatives: 6

Speedup: 4.0×

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: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x\_m \cdot \mathsf{fma}\left(x\_m, -0.37545125292247583, -0.00011824294398844343\right)\\ \mathbf{if}\;\left|x\_m\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \frac{x\_m \cdot \left(1.2732557730789702 - {t\_0}^{2}\right)}{1.128386358070218 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.7778892405807117}{x\_m}}{e^{{x\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* x_m (fma x_m -0.37545125292247583 -0.00011824294398844343))))
   (if (<= (fabs x_m) 5e-5)
     (+
      1e-9
      (/
       (* x_m (- 1.2732557730789702 (pow t_0 2.0)))
       (- 1.128386358070218 t_0)))
     (- 1.0 (/ (/ 0.7778892405807117 x_m) (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = x_m * fma(x_m, -0.37545125292247583, -0.00011824294398844343);
	double tmp;
	if (fabs(x_m) <= 5e-5) {
		tmp = 1e-9 + ((x_m * (1.2732557730789702 - pow(t_0, 2.0))) / (1.128386358070218 - t_0));
	} else {
		tmp = 1.0 - ((0.7778892405807117 / x_m) / exp(pow(x_m, 2.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(x_m * fma(x_m, -0.37545125292247583, -0.00011824294398844343))
	tmp = 0.0
	if (abs(x_m) <= 5e-5)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(1.2732557730789702 - (t_0 ^ 2.0))) / Float64(1.128386358070218 - t_0)));
	else
		tmp = Float64(1.0 - Float64(Float64(0.7778892405807117 / x_m) / exp((x_m ^ 2.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * -0.37545125292247583 + -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 5e-5], N[(1e-9 + N[(N[(x$95$m * N[(1.2732557730789702 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.128386358070218 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.7778892405807117 / x$95$m), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000024e-5

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in x around inf 54.5%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 56.3%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 96.4%

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

      1. *-commutative 96.4%

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

      2. flip-+ 96.4%

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

      3. associate-*l/ 96.4%

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

      4. metadata-eval 96.4%

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

      5. pow2 96.4%

         \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - \color{blue}{{\left(x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)}^{2}}\right) \cdot x}{1.128386358070218 - x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)} \]

      6. *-commutative 96.4%

         \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - {\left(x \cdot \left(\color{blue}{x \cdot -0.37545125292247583} - 0.00011824294398844343\right)\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)} \]

      7. fma-neg 96.4%

         \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - {\left(x \cdot \color{blue}{\mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)}\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)} \]

      8. metadata-eval 96.4%

         \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - {\left(x \cdot \mathsf{fma}\left(x, -0.37545125292247583, \color{blue}{-0.00011824294398844343}\right)\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)} \]

      9. *-commutative 96.4%

         \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - {\left(x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \left(\color{blue}{x \cdot -0.37545125292247583} - 0.00011824294398844343\right)} \]

      10. fma-neg 96.4%

          \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - {\left(x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \color{blue}{\mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)}} \]

      11. metadata-eval 96.4%

          \[\leadsto 10^{-9} + \frac{\left(1.2732557730789702 - {\left(x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \mathsf{fma}\left(x, -0.37545125292247583, \color{blue}{-0.00011824294398844343}\right)} \]

   10. Applied egg-rr 96.4%

       \[\leadsto 10^{-9} + \color{blue}{\frac{\left(1.2732557730789702 - {\left(x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)\right)}^{2}\right) \cdot x}{1.128386358070218 - x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)}} \]

   if 5.00000000000000024e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \frac{x \cdot \left(1.2732557730789702 - {\left(x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)\right)}^{2}\right)}{1.128386358070218 - x \cdot \mathsf{fma}\left(x, -0.37545125292247583, -0.00011824294398844343\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(x\_m \cdot -0.37545125292247583 - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.7778892405807117}{x\_m}}{e^{{x\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05)
   (+
    1e-9
    (*
     x_m
     (+
      1.128386358070218
      (* x_m (- (* x_m -0.37545125292247583) 0.00011824294398844343)))))
   (- 1.0 (/ (/ 0.7778892405807117 x_m) (exp (pow x_m 2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))));
	} else {
		tmp = 1.0 - ((0.7778892405807117 / x_m) / exp(pow(x_m, 2.0)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.05d0) then
        tmp = 1d-9 + (x_m * (1.128386358070218d0 + (x_m * ((x_m * (-0.37545125292247583d0)) - 0.00011824294398844343d0))))
    else
        tmp = 1.0d0 - ((0.7778892405807117d0 / x_m) / exp((x_m ** 2.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))));
	} else {
		tmp = 1.0 - ((0.7778892405807117 / x_m) / Math.exp(Math.pow(x_m, 2.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.05:
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))))
	else:
		tmp = 1.0 - ((0.7778892405807117 / x_m) / math.exp(math.pow(x_m, 2.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(x_m * -0.37545125292247583) - 0.00011824294398844343)))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.7778892405807117 / x_m) / exp((x_m ^ 2.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.05)
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))));
	else
		tmp = 1.0 - ((0.7778892405807117 / x_m) / exp((x_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.05], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(x$95$m * -0.37545125292247583), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.7778892405807117 / x$95$m), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in x around inf 69.2%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.4%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 66.6%

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

   if 1.05000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(x \cdot -0.37545125292247583 - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.7778892405807117}{x}}{e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.8% accurate, 47.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.72:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(x\_m \cdot -0.37545125292247583 - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.72)
   (+
    1e-9
    (*
     x_m
     (+
      1.128386358070218
      (* x_m (- (* x_m -0.37545125292247583) 0.00011824294398844343)))))
   1e-9))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.72) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.72d0) then
        tmp = 1d-9 + (x_m * (1.128386358070218d0 + (x_m * ((x_m * (-0.37545125292247583d0)) - 0.00011824294398844343d0))))
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.72) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.72:
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))))
	else:
		tmp = 1e-9
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.72)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(x_m * -0.37545125292247583) - 0.00011824294398844343)))));
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.72)
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((x_m * -0.37545125292247583) - 0.00011824294398844343))));
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.72], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(x$95$m * -0.37545125292247583), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 1.71999999999999997

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in x around inf 69.2%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.4%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 66.6%

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

   if 1.71999999999999997 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   10. Simplified 0.9%

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

   11. Taylor expanded in x around 0 5.7%

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

   12. Taylor expanded in x around 0 11.1%

       \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.72:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(x \cdot -0.37545125292247583 - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.6% accurate, 61.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 9500:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 9500.0)
   (+ 1e-9 (* x_m (+ 1.128386358070218 (* x_m -0.00011824294398844343))))
   1e-9))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 9500.0) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * -0.00011824294398844343)));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 9500.0d0) then
        tmp = 1d-9 + (x_m * (1.128386358070218d0 + (x_m * (-0.00011824294398844343d0))))
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 9500.0) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * -0.00011824294398844343)));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 9500.0:
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * -0.00011824294398844343)))
	else:
		tmp = 1e-9
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 9500.0)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * -0.00011824294398844343))));
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 9500.0)
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * -0.00011824294398844343)));
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 9500.0], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9500

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in x around inf 69.2%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.4%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 65.5%

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

      1. *-commutative 65.5%

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

   10. Simplified 65.5%

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

   if 9500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   10. Simplified 0.9%

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

   11. Taylor expanded in x around 0 5.7%

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

   12. Taylor expanded in x around 0 11.1%

       \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9500:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.4% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 900000000:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 900000000.0) (+ 1e-9 (* x_m 1.128386358070218)) 1e-9))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 900000000.0d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 900000000.0:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1e-9
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 900000000.0)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 900000000.0], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9e8

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around inf 69.3%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 65.2%

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

      1. *-commutative 65.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in x around 0 65.3%

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

   if 9e8 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

   8. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   10. Simplified 0.9%

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

   11. Taylor expanded in x around 0 5.6%

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

   12. Taylor expanded in x around 0 11.1%

       \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 900000000:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.6% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in x around inf 76.2%

   \[\leadsto \color{blue}{1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

7. Simplified 77.1%

   \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}} \]

8. Taylor expanded in x around 0 50.9%

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

   1. *-commutative 50.9%

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

10. Simplified 50.9%

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

11. Taylor expanded in x around 0 52.0%

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

12. Taylor expanded in x around 0 55.4%

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

13. Final simplification 55.4%

    \[\leadsto 10^{-9} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))))))