Jmat.Real.erf

Percentage Accurate: 78.9% → 99.7%

Time: 8.5s

Alternatives: 7

Speedup: 3.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\_m\right|\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 1.16 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + 1.128386358070218 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_1 \cdot \left(0.254829592 + t\_1 \cdot \left(\mathsf{fma}\left(1.061405429, \frac{1}{{t\_0}^{3}}, \frac{1.421413741}{t\_0}\right) - \left(0.284496736 + \frac{1.453152027}{{t\_0}^{2}}\right)\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.3275911 (fabs x_m)))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 1.16e-6)
     (+ 1e-9 (* 1.128386358070218 x_m))
     (-
      1.0
      (*
       (*
        t_1
        (+
         0.254829592
         (*
          t_1
          (-
           (fma 1.061405429 (/ 1.0 (pow t_0 3.0)) (/ 1.421413741 t_0))
           (+ 0.284496736 (/ 1.453152027 (pow t_0 2.0)))))))
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * fabs(x_m));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 1.16e-6) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (fma(1.061405429, (1.0 / pow(t_0, 3.0)), (1.421413741 / t_0)) - (0.284496736 + (1.453152027 / pow(t_0, 2.0))))))) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x_m)))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 1.16e-6)
		tmp = Float64(1e-9 + Float64(1.128386358070218 * x_m));
	else
		tmp = Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(fma(1.061405429, Float64(1.0 / (t_0 ^ 3.0)), Float64(1.421413741 / t_0)) - Float64(0.284496736 + Float64(1.453152027 / (t_0 ^ 2.0))))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.16e-6], N[(1e-9 + N[(1.128386358070218 * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.421413741 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(0.284496736 + N[(1.453152027 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1599999999999999e-6

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   if 1.1599999999999999e-6 < x

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right) - \left(\frac{8890523}{31250000} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\mathsf{fma}\left(\frac{1061405429}{1000000000}, \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}, \frac{1421413741}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right) - \left(\frac{8890523}{31250000} + \frac{\frac{1453152027}{1000000000}}{\color{blue}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\mathsf{fma}\left(\frac{1061405429}{1000000000}, \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}, \frac{\frac{1421413741}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right) - \left(\frac{8890523}{31250000} + \frac{\frac{1453152027}{1000000000}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   10. Applied rewrites 78.9%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\_m\right|\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 1.56 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + 1.128386358070218 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_1 \cdot \left(0.254829592 + t\_1 \cdot \left(-0.284496736 + \frac{\left(1.421413741 + 1.061405429 \cdot \frac{1}{{t\_0}^{2}}\right) - 1.453152027 \cdot t\_1}{t\_0}\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.3275911 (fabs x_m)))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 1.56e-6)
     (+ 1e-9 (* 1.128386358070218 x_m))
     (-
      1.0
      (*
       (*
        t_1
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (/
            (-
             (+ 1.421413741 (* 1.061405429 (/ 1.0 (pow t_0 2.0))))
             (* 1.453152027 t_1))
            t_0)))))
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * fabs(x_m));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 1.56e-6) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / pow(t_0, 2.0)))) - (1.453152027 * t_1)) / t_0))))) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (0.3275911d0 * abs(x_m))
    t_1 = 1.0d0 / t_0
    if (x_m <= 1.56d-6) then
        tmp = 1d-9 + (1.128386358070218d0 * x_m)
    else
        tmp = 1.0d0 - ((t_1 * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (((1.421413741d0 + (1.061405429d0 * (1.0d0 / (t_0 ** 2.0d0)))) - (1.453152027d0 * t_1)) / t_0))))) * exp(-(abs(x_m) * abs(x_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * Math.abs(x_m));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 1.56e-6) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / Math.pow(t_0, 2.0)))) - (1.453152027 * t_1)) / t_0))))) * Math.exp(-(Math.abs(x_m) * Math.abs(x_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (0.3275911 * math.fabs(x_m))
	t_1 = 1.0 / t_0
	tmp = 0
	if x_m <= 1.56e-6:
		tmp = 1e-9 + (1.128386358070218 * x_m)
	else:
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / math.pow(t_0, 2.0)))) - (1.453152027 * t_1)) / t_0))))) * math.exp(-(math.fabs(x_m) * math.fabs(x_m))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x_m)))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 1.56e-6)
		tmp = Float64(1e-9 + Float64(1.128386358070218 * x_m));
	else
		tmp = Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(Float64(Float64(1.421413741 + Float64(1.061405429 * Float64(1.0 / (t_0 ^ 2.0)))) - Float64(1.453152027 * t_1)) / t_0))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (0.3275911 * abs(x_m));
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x_m <= 1.56e-6)
		tmp = 1e-9 + (1.128386358070218 * x_m);
	else
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / (t_0 ^ 2.0)))) - (1.453152027 * t_1)) / t_0))))) * exp(-(abs(x_m) * abs(x_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.56e-6], N[(1e-9 + N[(1.128386358070218 * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(N[(N[(1.421413741 + N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.453152027 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001e-6

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   if 1.5600000000000001e-6 < x

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \color{blue}{\frac{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. Applied rewrites 78.9%

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

Alternative 3: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + 1.128386358070218 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{e^{x\_m \cdot x\_m} \cdot \mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.45e-6)
   (+ 1e-9 (* 1.128386358070218 x_m))
   (-
    1.0
    (*
     (/
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
            (fma x_m 0.3275911 1.0)))
          (fma x_m 0.3275911 1.0)))
        (fma x_m 0.3275911 1.0)))
      (* (exp (* x_m x_m)) (fma x_m 0.3275911 1.0)))
     1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.45e-6) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (exp((x_m * x_m)) * fma(x_m, 0.3275911, 1.0))) * 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.45e-6)
		tmp = Float64(1e-9 + Float64(1.128386358070218 * x_m));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(exp(Float64(x_m * x_m)) * fma(x_m, 0.3275911, 1.0))) * 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.45e-6], N[(1e-9 + N[(1.128386358070218 * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4500000000000001e-6

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   if 1.4500000000000001e-6 < x

   1. Initial program 78.9%

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

   3. Applied rewrites 78.9%

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

Alternative 4: 99.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;10^{-9} + 1.128386358070218 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{-0.7778892405807117}{x\_m} \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (+ 1e-9 (* 1.128386358070218 x_m))
   (-
    1.0
    (* (/ -0.7778892405807117 x_m) (exp (- (* (fabs x_m) (fabs x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0 - ((-0.7778892405807117 / x_m) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = 1d-9 + (1.128386358070218d0 * x_m)
    else
        tmp = 1.0d0 - (((-0.7778892405807117d0) / x_m) * exp(-(abs(x_m) * abs(x_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0 - ((-0.7778892405807117 / x_m) * Math.exp(-(Math.abs(x_m) * Math.abs(x_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = 1e-9 + (1.128386358070218 * x_m)
	else:
		tmp = 1.0 - ((-0.7778892405807117 / x_m) * math.exp(-(math.fabs(x_m) * math.fabs(x_m))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(1e-9 + Float64(1.128386358070218 * x_m));
	else
		tmp = Float64(1.0 - Float64(Float64(-0.7778892405807117 / x_m) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = 1e-9 + (1.128386358070218 * x_m);
	else
		tmp = 1.0 - ((-0.7778892405807117 / x_m) * exp(-(abs(x_m) * abs(x_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(1e-9 + N[(1.128386358070218 * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(-0.7778892405807117 / x$95$m), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   if 1.1000000000000001 < x

   1. Initial program 78.9%

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

   3. Applied rewrites 77.6%

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

   4. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{\frac{-63707398}{81897775}}{x}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. Applied rewrites 51.9%

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

Alternative 5: 61.5% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   if 1 < x

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   8. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 1.128386358070218, 10^{-9}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 59.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, 1.128386358070218, 10^{-9}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.95e-5) 1e-9 (fma 1.0 1.128386358070218 1e-9)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.95e-5) {
		tmp = 1e-9;
	} else {
		tmp = fma(1.0, 1.128386358070218, 1e-9);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.95e-5)
		tmp = 1e-9;
	else
		tmp = fma(1.0, 1.128386358070218, 1e-9);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.95e-5], 1e-9, N[(1.0 * 1.128386358070218 + 1e-9), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9499999999999999e-5

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000}} \]

   6. Applied rewrites 53.7%

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

   if 2.9499999999999999e-5 < x

   1. Initial program 78.9%

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

   3. Applied rewrites 29.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]

   6. Applied rewrites 52.6%

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

   8. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 1.128386358070218, 10^{-9}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.7% accurate, 99.9× 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 =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    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 78.9%

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

3. Applied rewrites 29.3%

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

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{1000000000}} \]

6. Applied rewrites 53.7%

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

Reproduce

herbie shell --seed 2025159 
(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)))))))