Jmat.Real.erf

Percentage Accurate: 79.2% → 99.3%

Time: 22.5s

Alternatives: 5

Speedup: 279.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.0001:\\ \;\;\;\;\left(\left(x_m \cdot 1.128386358070218 + 10^{-9}\right) + {x_m}^{3} \cdot -0.37545125292247583\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0001)
   (+
    (+
     (+ (* x_m 1.128386358070218) 1e-9)
     (* (pow x_m 3.0) -0.37545125292247583))
    (* (pow x_m 2.0) -0.00011824294398844343))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0001) {
		tmp = (((x_m * 1.128386358070218) + 1e-9) + (pow(x_m, 3.0) * -0.37545125292247583)) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.0001d0) then
        tmp = (((x_m * 1.128386358070218d0) + 1d-9) + ((x_m ** 3.0d0) * (-0.37545125292247583d0))) + ((x_m ** 2.0d0) * (-0.00011824294398844343d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.0001) {
		tmp = (((x_m * 1.128386358070218) + 1e-9) + (Math.pow(x_m, 3.0) * -0.37545125292247583)) + (Math.pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.0001:
		tmp = (((x_m * 1.128386358070218) + 1e-9) + (math.pow(x_m, 3.0) * -0.37545125292247583)) + (math.pow(x_m, 2.0) * -0.00011824294398844343)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0001)
		tmp = Float64(Float64(Float64(Float64(x_m * 1.128386358070218) + 1e-9) + Float64((x_m ^ 3.0) * -0.37545125292247583)) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.0001)
		tmp = (((x_m * 1.128386358070218) + 1e-9) + ((x_m ^ 3.0) * -0.37545125292247583)) + ((x_m ^ 2.0) * -0.00011824294398844343);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0001], N[(N[(N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

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

   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%

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

   6. Applied egg-rr 57.0%

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

   7. Taylor expanded in x around 0 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-+r+ 94.6%

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

      3. associate-+l+ 94.6%

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

      4. *-commutative 94.6%

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

      5. fma-def 94.6%

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

      6. *-commutative 94.6%

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

      7. *-commutative 94.6%

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

      8. fma-def 94.6%

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

   9. Simplified 94.6%

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

       1. rem-cube-cbrt 97.5%

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

       2. +-commutative 97.5%

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

       3. fma-udef 97.5%

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

       4. *-commutative 97.5%

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

       5. associate-+r+ 97.5%

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

       6. *-commutative 97.5%

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

   11. Applied egg-rr 97.5%

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

       1. fma-udef 97.5%

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

   13. Applied egg-rr 97.5%

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

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

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto {\color{blue}{\left(1 - 0.2592964135269039 \cdot \frac{1}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto {\left(1 - \color{blue}{\frac{0.2592964135269039 \cdot 1}{x \cdot e^{{x}^{2}}}}\right)}^{3} \]

      2. metadata-eval 100.0%

         \[\leadsto {\left(1 - \frac{\color{blue}{0.2592964135269039}}{x \cdot e^{{x}^{2}}}\right)}^{3} \]

   9. Simplified 100.0%

      \[\leadsto {\color{blue}{\left(1 - \frac{0.2592964135269039}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left(\left(x \cdot 1.128386358070218 + 10^{-9}\right) + {x}^{3} \cdot -0.37545125292247583\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.89:\\ \;\;\;\;x_m \cdot 1.128386358070218 + \left(10^{-9} + {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.89)
   (+
    (* x_m 1.128386358070218)
    (+ 1e-9 (* (pow x_m 2.0) -0.00011824294398844343)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.89) {
		tmp = (x_m * 1.128386358070218) + (1e-9 + (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0;
	}
	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 <= 0.89d0) then
        tmp = (x_m * 1.128386358070218d0) + (1d-9 + ((x_m ** 2.0d0) * (-0.00011824294398844343d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.89) {
		tmp = (x_m * 1.128386358070218) + (1e-9 + (Math.pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.89:
		tmp = (x_m * 1.128386358070218) + (1e-9 + (math.pow(x_m, 2.0) * -0.00011824294398844343))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.89)
		tmp = Float64(Float64(x_m * 1.128386358070218) + Float64(1e-9 + Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.89)
		tmp = (x_m * 1.128386358070218) + (1e-9 + ((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.89], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + N[(1e-9 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

   6. Applied egg-rr 72.2%

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

   7. Taylor expanded in x around 0 61.3%

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

      1. rem-cube-cbrt 63.1%

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

      2. associate-+r+ 63.1%

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

      3. *-commutative 63.1%

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

      4. *-commutative 63.1%

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

   9. Applied egg-rr 63.1%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto {\color{blue}{\left(1 - 0.2592964135269039 \cdot \frac{1}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto {\left(1 - \color{blue}{\frac{0.2592964135269039 \cdot 1}{x \cdot e^{{x}^{2}}}}\right)}^{3} \]

      2. metadata-eval 100.0%

         \[\leadsto {\left(1 - \frac{\color{blue}{0.2592964135269039}}{x \cdot e^{{x}^{2}}}\right)}^{3} \]

   9. Simplified 100.0%

      \[\leadsto {\color{blue}{\left(1 - \frac{0.2592964135269039}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;x \cdot 1.128386358070218 + \left(10^{-9} + {x}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 121.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.89:\\ \;\;\;\;x_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.89) (+ (* x_m 1.128386358070218) 1e-9) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.89) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0;
	}
	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 <= 0.89d0) then
        tmp = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.89) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.89:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.89)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.89)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.89], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

   6. Applied egg-rr 72.2%

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

   7. Taylor expanded in x around 0 61.3%

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

      1. rem-cube-cbrt 63.1%

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

      2. associate-+r+ 63.1%

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

      3. *-commutative 63.1%

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

      4. *-commutative 63.1%

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

   9. Applied egg-rr 63.1%

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

   10. Taylor expanded in x around 0 63.2%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto {\color{blue}{\left(1 - 0.2592964135269039 \cdot \frac{1}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto {\left(1 - \color{blue}{\frac{0.2592964135269039 \cdot 1}{x \cdot e^{{x}^{2}}}}\right)}^{3} \]

      2. metadata-eval 100.0%

         \[\leadsto {\left(1 - \frac{\color{blue}{0.2592964135269039}}{x \cdot e^{{x}^{2}}}\right)}^{3} \]

   9. Simplified 100.0%

      \[\leadsto {\color{blue}{\left(1 - \frac{0.2592964135269039}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 279.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) 1e-9 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	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 <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

   6. Applied egg-rr 72.2%

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

   7. Taylor expanded in x around 0 61.3%

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

   8. Taylor expanded in x around 0 65.2%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto {\color{blue}{\left(1 - 0.2592964135269039 \cdot \frac{1}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto {\left(1 - \color{blue}{\frac{0.2592964135269039 \cdot 1}{x \cdot e^{{x}^{2}}}}\right)}^{3} \]

      2. metadata-eval 100.0%

         \[\leadsto {\left(1 - \frac{\color{blue}{0.2592964135269039}}{x \cdot e^{{x}^{2}}}\right)}^{3} \]

   9. Simplified 100.0%

      \[\leadsto {\color{blue}{\left(1 - \frac{0.2592964135269039}{x \cdot e^{{x}^{2}}}\right)}}^{3} \]

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 74.3%

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

Alternative 5: 52.9% 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 79.9%

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

3. Simplified 79.9%

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

6. Applied egg-rr 79.5%

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

7. Taylor expanded in x around 0 45.6%

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

8. Taylor expanded in x around 0 51.1%

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

9. Final simplification 51.1%

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

Reproduce

herbie shell --seed 2024013 
(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)))))))