Jmat.Real.erf

Percentage Accurate: 79.1% → 86.5%

Time: 12.7s

Alternatives: 12

Speedup: 1.0×

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

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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 86.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

6. Applied rewrites 80.2%

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

8. Applied rewrites 87.2%

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

9. Final simplification 87.2%

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

Alternative 2: 79.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

6. Applied rewrites 80.2%

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

7. Final simplification 80.2%

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

Alternative 3: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot 0.3275911 - -1\\ t_1 := -0.3275911 \cdot \left|x\right|\\ t_2 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_1 - 1} \cdot e^{\left(-x\right) \cdot x}\\ t_3 := 1 - t\_1\\ \frac{1 + {t\_2}^{3}}{1 + \left({\left(\frac{\frac{\left(\frac{1.421413741}{t\_3} + {t\_3}^{-3} \cdot 1.061405429\right) - \left({t\_3}^{-2} \cdot 1.453152027 + 0.284496736\right)}{t\_0} + 0.254829592}{t\_0 \cdot {\left(e^{x}\right)}^{x}}\right)}^{2} - t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* (fabs x) 0.3275911) -1.0))
        (t_1 (* -0.3275911 (fabs x)))
        (t_2
         (*
          (/
           (+
            (/
             (+
              (/
               (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
               t_0)
              -0.284496736)
             t_0)
            0.254829592)
           (- t_1 1.0))
          (exp (* (- x) x))))
        (t_3 (- 1.0 t_1)))
   (/
    (+ 1.0 (pow t_2 3.0))
    (+
     1.0
     (-
      (pow
       (/
        (+
         (/
          (-
           (+ (/ 1.421413741 t_3) (* (pow t_3 -3.0) 1.061405429))
           (+ (* (pow t_3 -2.0) 1.453152027) 0.284496736))
          t_0)
         0.254829592)
        (* t_0 (pow (exp x) x)))
       2.0)
      t_2)))))

C

double code(double x) {
	double t_0 = (fabs(x) * 0.3275911) - -1.0;
	double t_1 = -0.3275911 * fabs(x);
	double t_2 = (((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (t_1 - 1.0)) * exp((-x * x));
	double t_3 = 1.0 - t_1;
	return (1.0 + pow(t_2, 3.0)) / (1.0 + (pow(((((((1.421413741 / t_3) + (pow(t_3, -3.0) * 1.061405429)) - ((pow(t_3, -2.0) * 1.453152027) + 0.284496736)) / t_0) + 0.254829592) / (t_0 * pow(exp(x), x))), 2.0) - t_2));
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (abs(x) * 0.3275911d0) - (-1.0d0)
    t_1 = (-0.3275911d0) * abs(x)
    t_2 = (((((((((1.061405429d0 / t_0) - 1.453152027d0) / t_0) - (-1.421413741d0)) / t_0) + (-0.284496736d0)) / t_0) + 0.254829592d0) / (t_1 - 1.0d0)) * exp((-x * x))
    t_3 = 1.0d0 - t_1
    code = (1.0d0 + (t_2 ** 3.0d0)) / (1.0d0 + ((((((((1.421413741d0 / t_3) + ((t_3 ** (-3.0d0)) * 1.061405429d0)) - (((t_3 ** (-2.0d0)) * 1.453152027d0) + 0.284496736d0)) / t_0) + 0.254829592d0) / (t_0 * (exp(x) ** x))) ** 2.0d0) - t_2))
end function

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * 0.3275911) - -1.0;
	double t_1 = -0.3275911 * Math.abs(x);
	double t_2 = (((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (t_1 - 1.0)) * Math.exp((-x * x));
	double t_3 = 1.0 - t_1;
	return (1.0 + Math.pow(t_2, 3.0)) / (1.0 + (Math.pow(((((((1.421413741 / t_3) + (Math.pow(t_3, -3.0) * 1.061405429)) - ((Math.pow(t_3, -2.0) * 1.453152027) + 0.284496736)) / t_0) + 0.254829592) / (t_0 * Math.pow(Math.exp(x), x))), 2.0) - t_2));
}

Python

def code(x):
	t_0 = (math.fabs(x) * 0.3275911) - -1.0
	t_1 = -0.3275911 * math.fabs(x)
	t_2 = (((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (t_1 - 1.0)) * math.exp((-x * x))
	t_3 = 1.0 - t_1
	return (1.0 + math.pow(t_2, 3.0)) / (1.0 + (math.pow(((((((1.421413741 / t_3) + (math.pow(t_3, -3.0) * 1.061405429)) - ((math.pow(t_3, -2.0) * 1.453152027) + 0.284496736)) / t_0) + 0.254829592) / (t_0 * math.pow(math.exp(x), x))), 2.0) - t_2))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * 0.3275911) - -1.0)
	t_1 = Float64(-0.3275911 * abs(x))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / Float64(t_1 - 1.0)) * exp(Float64(Float64(-x) * x)))
	t_3 = Float64(1.0 - t_1)
	return Float64(Float64(1.0 + (t_2 ^ 3.0)) / Float64(1.0 + Float64((Float64(Float64(Float64(Float64(Float64(Float64(1.421413741 / t_3) + Float64((t_3 ^ -3.0) * 1.061405429)) - Float64(Float64((t_3 ^ -2.0) * 1.453152027) + 0.284496736)) / t_0) + 0.254829592) / Float64(t_0 * (exp(x) ^ x))) ^ 2.0) - t_2)))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * 0.3275911) - -1.0;
	t_1 = -0.3275911 * abs(x);
	t_2 = (((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (t_1 - 1.0)) * exp((-x * x));
	t_3 = 1.0 - t_1;
	tmp = (1.0 + (t_2 ^ 3.0)) / (1.0 + ((((((((1.421413741 / t_3) + ((t_3 ^ -3.0) * 1.061405429)) - (((t_3 ^ -2.0) * 1.453152027) + 0.284496736)) / t_0) + 0.254829592) / (t_0 * (exp(x) ^ x))) ^ 2.0) - t_2));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - t$95$1), $MachinePrecision]}, N[(N[(1.0 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(N[(N[(N[(N[(N[(1.421413741 / t$95$3), $MachinePrecision] + N[(N[Power[t$95$3, -3.0], $MachinePrecision] * 1.061405429), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[t$95$3, -2.0], $MachinePrecision] * 1.453152027), $MachinePrecision] + 0.284496736), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$0 * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.2%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

7. Applied rewrites 80.2%

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

8. Final simplification 80.2%

   \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 - -1} - 1.453152027}{\left|x\right| \cdot 0.3275911 - -1} - -1.421413741}{\left|x\right| \cdot 0.3275911 - -1} + -0.284496736}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{-0.3275911 \cdot \left|x\right| - 1} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\left(\frac{1.421413741}{1 - -0.3275911 \cdot \left|x\right|} + {\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-3} \cdot 1.061405429\right) - \left({\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-2} \cdot 1.453152027 + 0.284496736\right)}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{\left(\left|x\right| \cdot 0.3275911 - -1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{2} - \frac{\frac{\frac{\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 - -1} - 1.453152027}{\left|x\right| \cdot 0.3275911 - -1} - -1.421413741}{\left|x\right| \cdot 0.3275911 - -1} + -0.284496736}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{-0.3275911 \cdot \left|x\right| - 1} \cdot e^{\left(-x\right) \cdot x}\right)} \]
9. Add Preprocessing

Alternative 4: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot 0.3275911 - -1\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := \frac{t\_1}{-0.3275911 \cdot \left|x\right| - 1} \cdot e^{\left(-x\right) \cdot x}\\ \frac{1 + {t\_2}^{3}}{1 + \left({\left(\frac{t\_1}{t\_0 \cdot {\left(e^{x}\right)}^{x}}\right)}^{2} - t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* (fabs x) 0.3275911) -1.0))
        (t_1
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_2 (* (/ t_1 (- (* -0.3275911 (fabs x)) 1.0)) (exp (* (- x) x)))))
   (/
    (+ 1.0 (pow t_2 3.0))
    (+ 1.0 (- (pow (/ t_1 (* t_0 (pow (exp x) x))) 2.0) t_2)))))

C

double code(double x) {
	double t_0 = (fabs(x) * 0.3275911) - -1.0;
	double t_1 = (((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_2 = (t_1 / ((-0.3275911 * fabs(x)) - 1.0)) * exp((-x * x));
	return (1.0 + pow(t_2, 3.0)) / (1.0 + (pow((t_1 / (t_0 * pow(exp(x), x))), 2.0) - t_2));
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (abs(x) * 0.3275911d0) - (-1.0d0)
    t_1 = (((((((1.061405429d0 / t_0) - 1.453152027d0) / t_0) - (-1.421413741d0)) / t_0) + (-0.284496736d0)) / t_0) + 0.254829592d0
    t_2 = (t_1 / (((-0.3275911d0) * abs(x)) - 1.0d0)) * exp((-x * x))
    code = (1.0d0 + (t_2 ** 3.0d0)) / (1.0d0 + (((t_1 / (t_0 * (exp(x) ** x))) ** 2.0d0) - t_2))
end function

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * 0.3275911) - -1.0;
	double t_1 = (((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_2 = (t_1 / ((-0.3275911 * Math.abs(x)) - 1.0)) * Math.exp((-x * x));
	return (1.0 + Math.pow(t_2, 3.0)) / (1.0 + (Math.pow((t_1 / (t_0 * Math.pow(Math.exp(x), x))), 2.0) - t_2));
}

Python

def code(x):
	t_0 = (math.fabs(x) * 0.3275911) - -1.0
	t_1 = (((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592
	t_2 = (t_1 / ((-0.3275911 * math.fabs(x)) - 1.0)) * math.exp((-x * x))
	return (1.0 + math.pow(t_2, 3.0)) / (1.0 + (math.pow((t_1 / (t_0 * math.pow(math.exp(x), x))), 2.0) - t_2))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * 0.3275911) - -1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592)
	t_2 = Float64(Float64(t_1 / Float64(Float64(-0.3275911 * abs(x)) - 1.0)) * exp(Float64(Float64(-x) * x)))
	return Float64(Float64(1.0 + (t_2 ^ 3.0)) / Float64(1.0 + Float64((Float64(t_1 / Float64(t_0 * (exp(x) ^ x))) ^ 2.0) - t_2)))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * 0.3275911) - -1.0;
	t_1 = (((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	t_2 = (t_1 / ((-0.3275911 * abs(x)) - 1.0)) * exp((-x * x));
	tmp = (1.0 + (t_2 ^ 3.0)) / (1.0 + (((t_1 / (t_0 * (exp(x) ^ x))) ^ 2.0) - t_2));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[(N[(-0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$1 / N[(t$95$0 * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.2%

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

5. Final simplification 80.2%

   \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 - -1} - 1.453152027}{\left|x\right| \cdot 0.3275911 - -1} - -1.421413741}{\left|x\right| \cdot 0.3275911 - -1} + -0.284496736}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{-0.3275911 \cdot \left|x\right| - 1} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 - -1} - 1.453152027}{\left|x\right| \cdot 0.3275911 - -1} - -1.421413741}{\left|x\right| \cdot 0.3275911 - -1} + -0.284496736}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{\left(\left|x\right| \cdot 0.3275911 - -1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{2} - \frac{\frac{\frac{\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 - -1} - 1.453152027}{\left|x\right| \cdot 0.3275911 - -1} - -1.421413741}{\left|x\right| \cdot 0.3275911 - -1} + -0.284496736}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{-0.3275911 \cdot \left|x\right| - 1} \cdot e^{\left(-x\right) \cdot x}\right)} \]
6. Add Preprocessing

Alternative 5: 79.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 80.1

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

6. Applied rewrites 80.1%

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

7. Final simplification 80.1%

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

Alternative 6: 79.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

5. Final simplification 80.1%

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

Alternative 7: 79.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

6. Applied rewrites 80.1%

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

Alternative 8: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fabs(x) * 0.3275911;
	double t_1 = t_0 - -1.0;
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (0.254829592 + ((((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / (1.0 - (0.10731592879921 * (x * x)))) * (1.0 - t_0)))) * exp((-x * 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
    real(8) :: t_1
    t_0 = abs(x) * 0.3275911d0
    t_1 = t_0 - (-1.0d0)
    code = 1.0d0 - (((1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))) * (0.254829592d0 + ((((((((1.061405429d0 / t_1) - 1.453152027d0) / t_1) - (-1.421413741d0)) / t_1) + (-0.284496736d0)) / (1.0d0 - (0.10731592879921d0 * (x * x)))) * (1.0d0 - t_0)))) * exp((-x * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot \color{blue}{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]
6. Step-by-step derivation

   1. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|\color{blue}{x}\right|\right)}^{2}\right)} \]

   2. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \]

   3. pow2 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\color{blue}{\left(\left|x\right|\right)}}^{2}\right)} \]

   4. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|\color{blue}{x}\right|\right)}^{2}\right)} \]

   5. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|\color{blue}{x}\right|\right)}^{2}\right)} \]

   6. pow2 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\color{blue}{\left(\left|x\right|\right)}}^{2}\right)} \]

   7. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|\color{blue}{x}\right|\right)}^{2}\right)} \]

   8. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \]

   9. sqr-abs-rev N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\color{blue}{\left(\left|x\right|\right)}}^{2}\right)} \]

   10. distribute-lft-neg-out N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\color{blue}{{\left(\left|x\right|\right)}^{2}}\right)} \]

   11. lift-fabs.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \]

   12. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   13. lift-fabs.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   14. lift-fabs.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   15. sqr-abs-rev N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot x\right)} \]

   16. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({x}^{2}\right)} \]

   17. lower-exp.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left({x}^{2}\right)} \]

   18. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot x\right)} \]

   19. distribute-lft-neg-out N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\left(\mathsf{neg}\left(x\right)\right) \cdot x} \]

   20. lift-*.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)} \cdot \left(1 - \left|x\right| \cdot \frac{3275911}{10000000}\right)\right)\right) \cdot e^{\left(\mathsf{neg}\left(x\right)\right) \cdot x} \]

7. Applied rewrites 80.1%

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

Alternative 9: 79.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = (fabs(x) * 0.3275911) - -1.0;
	return 1.0 + (((-1.0 / (1.0 + (0.3275911 * fabs(x)))) * ((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592)) * exp((-x * 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 = (abs(x) * 0.3275911d0) - (-1.0d0)
    code = 1.0d0 + ((((-1.0d0) / (1.0d0 + (0.3275911d0 * abs(x)))) * ((((((((1.061405429d0 / t_0) - 1.453152027d0) / t_0) - (-1.421413741d0)) / t_0) + (-0.284496736d0)) / t_0) + 0.254829592d0)) * exp((-x * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

5. Final simplification 80.1%

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

Alternative 10: 79.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = (fabs(x) * 0.3275911) - -1.0;
	return 1.0 - ((((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp((-x * 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 = (abs(x) * 0.3275911d0) - (-1.0d0)
    code = 1.0d0 - ((((((((((1.061405429d0 / t_0) - 1.453152027d0) / t_0) - (-1.421413741d0)) / t_0) + (-0.284496736d0)) / t_0) + 0.254829592d0) / t_0) * exp((-x * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

Alternative 11: 77.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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 = (abs(x) * 0.3275911d0) - (-1.0d0)
    code = 1.0d0 - ((((((((((1.061405429d0 / t_0) - 1.453152027d0) / t_0) - (-1.421413741d0)) / t_0) / t_0) + ((-0.284496736d0) / t_0)) + 0.254829592d0) / t_0) * 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{31853699}{125000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 78.8%

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 - -1} - 1.453152027}{\left|x\right| \cdot 0.3275911 - -1} - -1.421413741}{\left|x\right| \cdot 0.3275911 - -1} + -0.284496736}{\left|x\right| \cdot 0.3275911 - -1} + 0.254829592}{\left|x\right| \cdot 0.3275911 - -1} \cdot \color{blue}{1} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 78.8

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

9. Applied rewrites 78.8%

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

Alternative 12: 77.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 = (abs(x) * 0.3275911d0) - (-1.0d0)
    code = 1.0d0 - ((((((((((1.061405429d0 / t_0) - 1.453152027d0) / t_0) - (-1.421413741d0)) / t_0) + (-0.284496736d0)) / t_0) + 0.254829592d0) / t_0) * 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{1453152027}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} - \frac{-1421413741}{1000000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{-8890523}{31250000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} + \frac{31853699}{125000000}}{\left|x\right| \cdot \frac{3275911}{10000000} - -1} \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 78.8%

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

Reproduce

herbie shell --seed 2025058 
(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)))))))