Jmat.Real.erf

Percentage Accurate: 79.2% → 86.6%

Time: 13.6s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 86.6% accurate, 0.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (+ (/ 1.061405429 (fma x_m 0.3275911 1.0)) -1.453152027)
                (fma x_m 0.3275911 1.0)))
              (fma x_m 0.3275911 1.0)))
            (fma x_m 0.3275911 1.0)))
          (* t_0 (pow (exp x_m) x_m))))
        (t_2 (+ (pow t_1 2.0) 1.0)))
   (/
    (- (pow t_2 -1.0) (/ (pow t_1 4.0) t_2))
    (fma
     (/
      (+
       (/
        (+
         (/
          (+
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0))
           1.421413741)
          (fma 0.3275911 x_m 1.0))
         -0.284496736)
        (fma 0.3275911 x_m 1.0))
       0.254829592)
      t_0)
     (exp (* (- x_m) x_m))
     1.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

6. Applied rewrites 79.4%

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

8. Applied rewrites 86.7%

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

Alternative 2: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\_m\right|\right)}^{-1}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(x\_m \cdot x\_m, -0.999999999, 0.999999999\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x_m))) -1.0)))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
         (exp (* (- x_m) x_m)))
        0.0)
     1.0
     (- 1.0 (fma (* x_m x_m) -0.999999999 0.999999999)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x_m))), -1.0);
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - fma((x_m * x_m), -0.999999999, 0.999999999);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x_m))) ^ -1.0
	tmp = 0.0
	if (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(-x_m) * x_m))) <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - fma(Float64(x_m * x_m), -0.999999999, 0.999999999));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[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[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], 1.0, N[(1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.999999999 + 0.999999999), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 57.7%

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{999999999}{1000000000} + \frac{-564193179035109}{500000000000000} \cdot x\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{-564193179035109}{500000000000000} \cdot x + \frac{999999999}{1000000000}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      2. lower-fma.f64 57.2

         \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. Applied rewrites 57.2%

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \color{blue}{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \frac{999999999}{1000000000}} \]

      2. unpow2 N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}\right)} \cdot \frac{999999999}{1000000000} \]

      3. sqr-abs-rev N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \cdot \frac{999999999}{1000000000} \]

      4. unpow2 N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)} \cdot \frac{999999999}{1000000000} \]

      5. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot {x}^{2}}} \cdot \frac{999999999}{1000000000} \]

      6. lower-*.f64 N/A

         \[\leadsto 1 - \color{blue}{e^{-1 \cdot {x}^{2}} \cdot \frac{999999999}{1000000000}} \]

      7. lower-exp.f64 N/A

         \[\leadsto 1 - \color{blue}{e^{-1 \cdot {x}^{2}}} \cdot \frac{999999999}{1000000000} \]

      8. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left({x}^{2}\right)}} \cdot \frac{999999999}{1000000000} \]

      9. unpow2 N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \cdot \frac{999999999}{1000000000} \]

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

          \[\leadsto 1 - e^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot x}} \cdot \frac{999999999}{1000000000} \]

      11. lower-*.f64 N/A

          \[\leadsto 1 - e^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot x}} \cdot \frac{999999999}{1000000000} \]

      12. lower-neg.f64 57.0

          \[\leadsto 1 - e^{\color{blue}{\left(-x\right)} \cdot x} \cdot 0.999999999 \]

   10. Applied rewrites 57.0%

       \[\leadsto 1 - \color{blue}{e^{\left(-x\right) \cdot x} \cdot 0.999999999} \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 - \left(\frac{999999999}{1000000000} + \color{blue}{\frac{-999999999}{1000000000} \cdot {x}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 57.0%

       \[\leadsto 1 - \mathsf{fma}\left(x \cdot x, \color{blue}{-0.999999999}, 0.999999999\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(1.421413741 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-1.453152027 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(x \cdot x, -0.999999999, 0.999999999\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\_m\right|\right)}^{-1}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - 0.999999999\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x_m))) -1.0)))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
         (exp (* (- x_m) x_m)))
        0.98)
     1.0
     (- 1.0 0.999999999))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x_m))), -1.0);
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m))) <= 0.98) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - 0.999999999;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (0.3275911d0 * abs(x_m))) ** (-1.0d0)
    if (((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp((-x_m * x_m))) <= 0.98d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 - 0.999999999d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.pow((1.0 + (0.3275911 * Math.abs(x_m))), -1.0);
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp((-x_m * x_m))) <= 0.98) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - 0.999999999;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.pow((1.0 + (0.3275911 * math.fabs(x_m))), -1.0)
	tmp = 0
	if ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp((-x_m * x_m))) <= 0.98:
		tmp = 1.0
	else:
		tmp = 1.0 - 0.999999999
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x_m))) ^ -1.0
	tmp = 0.0
	if (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(-x_m) * x_m))) <= 0.98)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - 0.999999999);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (1.0 + (0.3275911 * abs(x_m))) ^ -1.0;
	tmp = 0.0;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m))) <= 0.98)
		tmp = 1.0;
	else
		tmp = 1.0 - 0.999999999;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[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[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.98], 1.0, N[(1.0 - 0.999999999), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.97999999999999998

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.97999999999999998 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 57.7%

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{999999999}{1000000000} + \frac{-564193179035109}{500000000000000} \cdot x\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{-564193179035109}{500000000000000} \cdot x + \frac{999999999}{1000000000}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      2. lower-fma.f64 57.2

         \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. Applied rewrites 57.2%

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \color{blue}{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \frac{999999999}{1000000000}} \]

      2. unpow2 N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}\right)} \cdot \frac{999999999}{1000000000} \]

      3. sqr-abs-rev N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \cdot \frac{999999999}{1000000000} \]

      4. unpow2 N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)} \cdot \frac{999999999}{1000000000} \]

      5. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot {x}^{2}}} \cdot \frac{999999999}{1000000000} \]

      6. lower-*.f64 N/A

         \[\leadsto 1 - \color{blue}{e^{-1 \cdot {x}^{2}} \cdot \frac{999999999}{1000000000}} \]

      7. lower-exp.f64 N/A

         \[\leadsto 1 - \color{blue}{e^{-1 \cdot {x}^{2}}} \cdot \frac{999999999}{1000000000} \]

      8. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left({x}^{2}\right)}} \cdot \frac{999999999}{1000000000} \]

      9. unpow2 N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \cdot \frac{999999999}{1000000000} \]

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

          \[\leadsto 1 - e^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot x}} \cdot \frac{999999999}{1000000000} \]

      11. lower-*.f64 N/A

          \[\leadsto 1 - e^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot x}} \cdot \frac{999999999}{1000000000} \]

      12. lower-neg.f64 57.0

          \[\leadsto 1 - e^{\color{blue}{\left(-x\right)} \cdot x} \cdot 0.999999999 \]

   10. Applied rewrites 57.0%

       \[\leadsto 1 - \color{blue}{e^{\left(-x\right) \cdot x} \cdot 0.999999999} \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 - \frac{999999999}{1000000000} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.0%

       \[\leadsto 1 - 0.999999999 \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(1.421413741 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-1.453152027 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - 0.999999999\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ t_1 := \frac{0.999999998}{{t\_0}^{2}} + 1\\ \frac{{t\_1}^{-1} - \frac{0.999999996}{{t\_0}^{4} \cdot t\_1}}{\frac{0.999999999}{t\_0} + 1} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1 (+ (/ 0.999999998 (pow t_0 2.0)) 1.0)))
   (/
    (- (pow t_1 -1.0) (/ 0.999999996 (* (pow t_0 4.0) t_1)))
    (+ (/ 0.999999999 t_0) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = (0.999999998 / pow(t_0, 2.0)) + 1.0;
	return (pow(t_1, -1.0) - (0.999999996 / (pow(t_0, 4.0) * t_1))) / ((0.999999999 / t_0) + 1.0);
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = Float64(Float64(0.999999998 / (t_0 ^ 2.0)) + 1.0)
	return Float64(Float64((t_1 ^ -1.0) - Float64(0.999999996 / Float64((t_0 ^ 4.0) * t_1))) / Float64(Float64(0.999999999 / t_0) + 1.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.999999998 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[Power[t$95$1, -1.0], $MachinePrecision] - N[(0.999999996 / N[(N[Power[t$95$0, 4.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.999999999 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

6. Applied rewrites 79.4%

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

8. Applied rewrites 86.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{1 + \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}} - \frac{999999996000000005999999996000000001}{1000000000000000000000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(1 + \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{1 + \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}} - \frac{999999996000000005999999996000000001}{1000000000000000000000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(1 + \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]

11. Applied rewrites 86.0%

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{0.999999998}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} + 1} - \frac{0.999999996}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4} \cdot \left(\frac{0.999999998}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} + 1\right)}}{\frac{0.999999999}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1}} \]

12. Final simplification 86.0%

    \[\leadsto \frac{{\left(\frac{0.999999998}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} + 1\right)}^{-1} - \frac{0.999999996}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4} \cdot \left(\frac{0.999999998}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} + 1\right)}}{\frac{0.999999999}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1} \]
13. Add Preprocessing

Alternative 5: 78.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \frac{1 - \frac{0.999999997}{{t\_0}^{3}}}{\mathsf{fma}\left(\frac{\frac{0.999999999}{t\_0} + 1}{t\_0}, 0.999999999, 1\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (/
    (- 1.0 (/ 0.999999997 (pow t_0 3.0)))
    (fma (/ (+ (/ 0.999999999 t_0) 1.0) t_0) 0.999999999 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	return (1.0 - (0.999999997 / pow(t_0, 3.0))) / fma((((0.999999999 / t_0) + 1.0) / t_0), 0.999999999, 1.0);
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(Float64(1.0 - Float64(0.999999997 / (t_0 ^ 3.0))) / fma(Float64(Float64(Float64(0.999999999 / t_0) + 1.0) / t_0), 0.999999999, 1.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(N[(1.0 - N[(0.999999997 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.999999999 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.999999999 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

6. Applied rewrites 79.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 - \frac{999999997000000002999999999}{1000000000000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \frac{999999997000000002999999999}{1000000000000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \frac{999999997000000002999999999}{1000000000000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{999999997000000002999999999}{1000000000000000000000000000} \cdot 1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 - \frac{\color{blue}{\frac{999999997000000002999999999}{1000000000000000000000000000}}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1 - \frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{\color{blue}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1 - \frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{{\color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right| + 1\right)}}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1 - \frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{{\left(\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1\right)}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{{\color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{{\left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{3275911}{10000000}, 1\right)\right)}^{3}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   11. +-commutative N/A

       \[\leadsto \frac{1 - \frac{\frac{999999997000000002999999999}{1000000000000000000000000000}}{{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}^{3}}}{\color{blue}{\frac{999999999}{1000000000} \cdot \frac{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + 1}} \]

9. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.999999997}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{3}}}{\mathsf{fma}\left(\frac{\frac{0.999999999}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 0.999999999, 1\right)}} \]
10. Add Preprocessing

Alternative 6: 78.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - \mathsf{fma}\left(-1.128386358070218, x\_m, 0.999999999\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (- 1.0 (* (fma -1.128386358070218 x_m 0.999999999) (exp (* (- x_m) x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - (fma(-1.128386358070218, x_m, 0.999999999) * exp((-x_m * x_m)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - Float64(fma(-1.128386358070218, x_m, 0.999999999) * exp(Float64(Float64(-x_m) * x_m))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[(N[(-1.128386358070218 * x$95$m + 0.999999999), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 - \color{blue}{\left(\frac{999999999}{1000000000} + \frac{-564193179035109}{500000000000000} \cdot x\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 - \color{blue}{\left(\frac{-564193179035109}{500000000000000} \cdot x + \frac{999999999}{1000000000}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. lower-fma.f64 79.3

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

7. Applied rewrites 79.3%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \color{blue}{e^{-\left|x\right| \cdot \left|x\right|}} \]

   2. lift-neg.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)}} \]

   3. exp-neg N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \color{blue}{\frac{1}{e^{\left|x\right| \cdot \left|x\right|}}} \]

   4. lift-*.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \frac{1}{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}} \]

   5. lift-fabs.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \frac{1}{e^{\color{blue}{\left|x\right|} \cdot \left|x\right|}} \]

   6. lift-fabs.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \frac{1}{e^{\left|x\right| \cdot \color{blue}{\left|x\right|}}} \]

   7. sqr-abs N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \frac{1}{e^{\color{blue}{x \cdot x}}} \]

   8. pow-exp N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \frac{1}{\color{blue}{{\left(e^{x}\right)}^{x}}} \]

   9. lift-exp.f64 N/A

      \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \frac{1}{{\color{blue}{\left(e^{x}\right)}}^{x}} \]

   10. pow-flip N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \color{blue}{{\left(e^{x}\right)}^{\left(\mathsf{neg}\left(x\right)\right)}} \]

   11. lift-exp.f64 N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot {\color{blue}{\left(e^{x}\right)}}^{\left(\mathsf{neg}\left(x\right)\right)} \]

   12. lift-neg.f64 N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot {\left(e^{x}\right)}^{\color{blue}{\left(-x\right)}} \]

   13. exp-prod N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot \color{blue}{e^{x \cdot \left(-x\right)}} \]

   14. *-commutative N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot e^{\color{blue}{\left(-x\right) \cdot x}} \]

   15. lift-*.f64 N/A

       \[\leadsto 1 - \mathsf{fma}\left(\frac{-564193179035109}{500000000000000}, x, \frac{999999999}{1000000000}\right) \cdot e^{\color{blue}{\left(-x\right) \cdot x}} \]

   16. lift-exp.f64 79.3

       \[\leadsto 1 - \mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right) \cdot \color{blue}{e^{\left(-x\right) \cdot x}} \]

9. Applied rewrites 79.3%

   \[\leadsto 1 - \mathsf{fma}\left(-1.128386358070218, x, 0.999999999\right) \cdot \color{blue}{e^{\left(-x\right) \cdot x}} \]
10. Add Preprocessing

Alternative 7: 78.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - e^{\left(-x\_m\right) \cdot x\_m} \cdot 0.999999999 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (- 1.0 (* (exp (* (- x_m) x_m)) 0.999999999)))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - (exp((-x_m * x_m)) * 0.999999999);
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1.0d0 - (exp((-x_m * x_m)) * 0.999999999d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 - (Math.exp((-x_m * x_m)) * 0.999999999);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 - (math.exp((-x_m * x_m)) * 0.999999999)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - Float64(exp(Float64(Float64(-x_m) * x_m)) * 0.999999999))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 - (exp((-x_m * x_m)) * 0.999999999);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[(N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] * 0.999999999), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 - \color{blue}{\frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto 1 - \color{blue}{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \frac{999999999}{1000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 - \color{blue}{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \frac{999999999}{1000000000}} \]

   3. lower-exp.f64 N/A

      \[\leadsto 1 - \color{blue}{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \cdot \frac{999999999}{1000000000} \]

   4. unpow2 N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}\right)} \cdot \frac{999999999}{1000000000} \]

   5. sqr-abs-rev N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \cdot \frac{999999999}{1000000000} \]

   6. unpow2 N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)} \cdot \frac{999999999}{1000000000} \]

   7. lower-neg.f64 N/A

      \[\leadsto 1 - e^{\color{blue}{-{x}^{2}}} \cdot \frac{999999999}{1000000000} \]

   8. unpow2 N/A

      \[\leadsto 1 - e^{-\color{blue}{x \cdot x}} \cdot \frac{999999999}{1000000000} \]

   9. lower-*.f64 79.2

      \[\leadsto 1 - e^{-\color{blue}{x \cdot x}} \cdot 0.999999999 \]

7. Applied rewrites 79.2%

   \[\leadsto 1 - \color{blue}{e^{-x \cdot x} \cdot 0.999999999} \]

8. Final simplification 79.2%

   \[\leadsto 1 - e^{\left(-x\right) \cdot x} \cdot 0.999999999 \]
9. Add Preprocessing

Alternative 8: 76.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \frac{1 - \frac{\frac{0.999999998}{t\_0}}{t\_0}}{\frac{0.999999999}{t\_0} + 1} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (/ (- 1.0 (/ (/ 0.999999998 t_0) t_0)) (+ (/ 0.999999999 t_0) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	return (1.0 - ((0.999999998 / t_0) / t_0)) / ((0.999999999 / t_0) + 1.0);
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(Float64(1.0 - Float64(Float64(0.999999998 / t_0) / t_0)) / Float64(Float64(0.999999999 / t_0) + 1.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(N[(1.0 - N[(N[(0.999999998 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.999999999 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

6. Applied rewrites 79.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 - \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{999999998000000001}{1000000000000000000} \cdot 1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 - \frac{\color{blue}{\frac{999999998000000001}{1000000000000000000}}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{999999998000000001}{1000000000000000000}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{\color{blue}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right| + 1\right)}}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{3275911}{10000000}, 1\right)\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   11. +-commutative N/A

       \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}^{2}}}{\color{blue}{\frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + 1}} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}^{2}}}{\color{blue}{\frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + 1}} \]

9. Applied rewrites 78.7%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.999999998}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}}{\frac{0.999999999}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1}} \]
10. Step-by-step derivation

11. Applied rewrites 78.7%

    \[\leadsto \frac{1 - \frac{\frac{0.999999998}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\frac{0.999999999}{\color{blue}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} + 1} \]
12. Add Preprocessing

Alternative 9: 76.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - \frac{0.999999999}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (- 1.0 (/ 0.999999999 (fma (fabs x_m) 0.3275911 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - (0.999999999 / fma(fabs(x_m), 0.3275911, 1.0));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - Float64(0.999999999 / fma(abs(x_m), 0.3275911, 1.0)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[(0.999999999 / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

6. Applied rewrites 79.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 - \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \frac{999999998000000001}{1000000000000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{999999998000000001}{1000000000000000000} \cdot 1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 - \frac{\color{blue}{\frac{999999998000000001}{1000000000000000000}}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{999999998000000001}{1000000000000000000}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{\color{blue}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right| + 1\right)}}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{3275911}{10000000}, 1\right)\right)}^{2}}}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

   11. +-commutative N/A

       \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}^{2}}}{\color{blue}{\frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + 1}} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{999999998000000001}{1000000000000000000}}{{\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}^{2}}}{\color{blue}{\frac{999999999}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + 1}} \]

9. Applied rewrites 78.7%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.999999998}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}}{\frac{0.999999999}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1}} \]
10. Step-by-step derivation

11. Applied rewrites 78.7%

    \[\leadsto \color{blue}{1 - \frac{0.999999999}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \]
12. Add Preprocessing

Alternative 10: 55.6% accurate, 262.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 79.5%

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

4. Applied rewrites 79.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 56.9%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(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)))))))