Jmat.Real.erf

Percentage Accurate: 79.2% → 83.4%

Time: 9.0s

Alternatives: 15

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: 79.2% 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: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := e^{x \cdot x}\\ t_3 := \frac{t\_1}{t\_0 \cdot t\_2}\\ t_4 := \mathsf{fma}\left(t\_3, t\_3 + 1, 1\right)\\ t_5 := \frac{t\_1}{t\_2 \cdot t\_0}\\ t_6 := \left(1 + {t\_5}^{6}\right) + {t\_5}^{3}\\ \frac{\frac{1}{t\_6}}{t\_4} - \frac{\frac{{t\_5}^{9}}{t\_6}}{t\_4} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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 (exp (* x x)))
        (t_3 (/ t_1 (* t_0 t_2)))
        (t_4 (fma t_3 (+ t_3 1.0) 1.0))
        (t_5 (/ t_1 (* t_2 t_0)))
        (t_6 (+ (+ 1.0 (pow t_5 6.0)) (pow t_5 3.0))))
   (- (/ (/ 1.0 t_6) t_4) (/ (/ (pow t_5 9.0) t_6) t_4))))

C

double code(double x) {
	double t_0 = fma(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 = exp((x * x));
	double t_3 = t_1 / (t_0 * t_2);
	double t_4 = fma(t_3, (t_3 + 1.0), 1.0);
	double t_5 = t_1 / (t_2 * t_0);
	double t_6 = (1.0 + pow(t_5, 6.0)) + pow(t_5, 3.0);
	return ((1.0 / t_6) / t_4) - ((pow(t_5, 9.0) / t_6) / t_4);
}

Julia

function code(x)
	t_0 = fma(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 = exp(Float64(x * x))
	t_3 = Float64(t_1 / Float64(t_0 * t_2))
	t_4 = fma(t_3, Float64(t_3 + 1.0), 1.0)
	t_5 = Float64(t_1 / Float64(t_2 * t_0))
	t_6 = Float64(Float64(1.0 + (t_5 ^ 6.0)) + (t_5 ^ 3.0))
	return Float64(Float64(Float64(1.0 / t_6) / t_4) - Float64(Float64((t_5 ^ 9.0) / t_6) / t_4))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 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[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(t$95$3 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(1.0 + N[Power[t$95$5, 6.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$6), $MachinePrecision] / t$95$4), $MachinePrecision] - N[(N[(N[Power[t$95$5, 9.0], $MachinePrecision] / t$95$6), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.3%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

7. Applied rewrites 80.4%

   \[\leadsto \frac{\color{blue}{\frac{1}{\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

9. Applied rewrites 83.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} - \frac{\frac{{\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)}} \]
10. Add Preprocessing

Alternative 2: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := e^{x \cdot x}\\ t_3 := \frac{t\_1}{t\_0 \cdot t\_2}\\ t_4 := \frac{t\_1}{t\_2 \cdot t\_0}\\ t_5 := \left(1 + {t\_4}^{6}\right) + {t\_4}^{3}\\ \frac{\frac{1}{t\_5} - \frac{{t\_4}^{9}}{t\_5}}{\mathsf{fma}\left(t\_3, t\_3 + 1, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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 (exp (* x x)))
        (t_3 (/ t_1 (* t_0 t_2)))
        (t_4 (/ t_1 (* t_2 t_0)))
        (t_5 (+ (+ 1.0 (pow t_4 6.0)) (pow t_4 3.0))))
   (/ (- (/ 1.0 t_5) (/ (pow t_4 9.0) t_5)) (fma t_3 (+ t_3 1.0) 1.0))))

C

double code(double x) {
	double t_0 = fma(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 = exp((x * x));
	double t_3 = t_1 / (t_0 * t_2);
	double t_4 = t_1 / (t_2 * t_0);
	double t_5 = (1.0 + pow(t_4, 6.0)) + pow(t_4, 3.0);
	return ((1.0 / t_5) - (pow(t_4, 9.0) / t_5)) / fma(t_3, (t_3 + 1.0), 1.0);
}

Julia

function code(x)
	t_0 = fma(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 = exp(Float64(x * x))
	t_3 = Float64(t_1 / Float64(t_0 * t_2))
	t_4 = Float64(t_1 / Float64(t_2 * t_0))
	t_5 = Float64(Float64(1.0 + (t_4 ^ 6.0)) + (t_4 ^ 3.0))
	return Float64(Float64(Float64(1.0 / t_5) - Float64((t_4 ^ 9.0) / t_5)) / fma(t_3, Float64(t_3 + 1.0), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 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[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(1.0 + N[Power[t$95$4, 6.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$5), $MachinePrecision] - N[(N[Power[t$95$4, 9.0], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(t$95$3 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.3%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

7. Applied rewrites 80.4%

   \[\leadsto \frac{\color{blue}{\frac{1}{\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]
8. Add Preprocessing

Alternative 3: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - -0.3275911 \cdot \left|x\right|\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := e^{x \cdot x}\\ t_3 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_4 := \frac{\frac{\frac{\left(1.421413741 + \frac{1.061405429}{t\_0 \cdot t\_0}\right) - \frac{1.453152027}{t\_0}}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_2 \cdot t\_1}\\ t_5 := {t\_4}^{3}\\ t_6 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_3} - 1.453152027}{t\_3} - -1.421413741}{t\_3} + -0.284496736}{t\_3} + 0.254829592}{t\_3 \cdot t\_2}\\ \frac{\frac{1 - {t\_4}^{9}}{1 + \mathsf{fma}\left(t\_5, t\_5, 1 \cdot t\_5\right)}}{\mathsf{fma}\left(t\_6, t\_6 + 1, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* -0.3275911 (fabs x))))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (exp (* x x)))
        (t_3 (fma (fabs x) 0.3275911 1.0))
        (t_4
         (/
          (+
           (/
            (+
             (/
              (-
               (+ 1.421413741 (/ 1.061405429 (* t_0 t_0)))
               (/ 1.453152027 t_0))
              t_1)
             -0.284496736)
            t_1)
           0.254829592)
          (* t_2 t_1)))
        (t_5 (pow t_4 3.0))
        (t_6
         (/
          (+
           (/
            (+
             (/
              (- (/ (- (/ 1.061405429 t_3) 1.453152027) t_3) -1.421413741)
              t_3)
             -0.284496736)
            t_3)
           0.254829592)
          (* t_3 t_2))))
   (/
    (/ (- 1.0 (pow t_4 9.0)) (+ 1.0 (fma t_5 t_5 (* 1.0 t_5))))
    (fma t_6 (+ t_6 1.0) 1.0))))

C

double code(double x) {
	double t_0 = 1.0 - (-0.3275911 * fabs(x));
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = exp((x * x));
	double t_3 = fma(fabs(x), 0.3275911, 1.0);
	double t_4 = ((((((1.421413741 + (1.061405429 / (t_0 * t_0))) - (1.453152027 / t_0)) / t_1) + -0.284496736) / t_1) + 0.254829592) / (t_2 * t_1);
	double t_5 = pow(t_4, 3.0);
	double t_6 = ((((((((1.061405429 / t_3) - 1.453152027) / t_3) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592) / (t_3 * t_2);
	return ((1.0 - pow(t_4, 9.0)) / (1.0 + fma(t_5, t_5, (1.0 * t_5)))) / fma(t_6, (t_6 + 1.0), 1.0);
}

Julia

function code(x)
	t_0 = Float64(1.0 - Float64(-0.3275911 * abs(x)))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = exp(Float64(x * x))
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.421413741 + Float64(1.061405429 / Float64(t_0 * t_0))) - Float64(1.453152027 / t_0)) / t_1) + -0.284496736) / t_1) + 0.254829592) / Float64(t_2 * t_1))
	t_5 = t_4 ^ 3.0
	t_6 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_3) - 1.453152027) / t_3) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592) / Float64(t_3 * t_2))
	return Float64(Float64(Float64(1.0 - (t_4 ^ 9.0)) / Float64(1.0 + fma(t_5, t_5, Float64(1.0 * t_5)))) / fma(t_6, Float64(t_6 + 1.0), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[(-0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(1.421413741 + N[(1.061405429 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.453152027 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$4, 3.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$3), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$3), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$4, 9.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$5 * t$95$5 + N[(1.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 * N[(t$95$6 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.3%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \color{blue}{\frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

8. Applied rewrites 79.3%

   \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\color{blue}{\left(1.421413741 + \frac{1.061405429}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)}\right) - \frac{1.453152027}{1 - -0.3275911 \cdot \left|x\right|}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]
10. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \color{blue}{\frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

11. Applied rewrites 79.3%

    \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(1.421413741 + \frac{1.061405429}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)}\right) - \frac{1.453152027}{1 - -0.3275911 \cdot \left|x\right|}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\color{blue}{\left(1.421413741 + \frac{1.061405429}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)}\right) - \frac{1.453152027}{1 - -0.3275911 \cdot \left|x\right|}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]
13. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \color{blue}{\frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

14. Applied rewrites 79.3%

    \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(1.421413741 + \frac{1.061405429}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)}\right) - \frac{1.453152027}{1 - -0.3275911 \cdot \left|x\right|}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\left(1.421413741 + \frac{1.061405429}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)}\right) - \frac{1.453152027}{1 - -0.3275911 \cdot \left|x\right|}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\color{blue}{\left(1.421413741 + \frac{1.061405429}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)}\right) - \frac{1.453152027}{1 - -0.3275911 \cdot \left|x\right|}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

15. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]
16. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}\right) - \frac{\frac{1453152027}{1000000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \color{blue}{\frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

17. Applied rewrites 79.3%

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

Alternative 4: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{e^{x \cdot x} \cdot t\_0}\\ \frac{1 - {t\_1}^{9}}{\left(\left(1 + {t\_1}^{6}\right) + {t\_1}^{3}\right) \cdot \mathsf{fma}\left(t\_1 + 1, t\_1, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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)
          (* (exp (* x x)) t_0))))
   (/
    (- 1.0 (pow t_1 9.0))
    (* (+ (+ 1.0 (pow t_1 6.0)) (pow t_1 3.0)) (fma (+ t_1 1.0) t_1 1.0)))))

C

double code(double x) {
	double t_0 = fma(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) / (exp((x * x)) * t_0);
	return (1.0 - pow(t_1, 9.0)) / (((1.0 + pow(t_1, 6.0)) + pow(t_1, 3.0)) * fma((t_1 + 1.0), t_1, 1.0));
}

Julia

function code(x)
	t_0 = fma(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(exp(Float64(x * x)) * t_0))
	return Float64(Float64(1.0 - (t_1 ^ 9.0)) / Float64(Float64(Float64(1.0 + (t_1 ^ 6.0)) + (t_1 ^ 3.0)) * fma(Float64(t_1 + 1.0), t_1, 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 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[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$1, 9.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[Power[t$95$1, 6.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 + 1.0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.3%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}} + 1, 1\right)} \]

7. Applied rewrites 79.3%

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(\left(1 + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}\right) \cdot \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)}} \]
8. Add Preprocessing

Alternative 5: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((((((((((1.061405429 / t_0) - 1.453152027) / t_0) / t_0) - (-1.421413741 / t_0)) + -0.284496736) / fma(0.3275911, fabs(x), 1.0)) + 0.254829592) / fma(-0.10731592879921, (x * x), 1.0)) * ((1.0 - (0.3275911 * fabs(x))) * exp(-(x * x))));
}

Julia

function code(x)
	t_0 = fma(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) / t_0) - Float64(-1.421413741 / t_0)) + -0.284496736) / fma(0.3275911, abs(x), 1.0)) + 0.254829592) / fma(-0.10731592879921, Float64(x * x), 1.0)) * Float64(Float64(1.0 - Float64(0.3275911 * abs(x))) * exp(Float64(-Float64(x * x))))))
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

   \[\leadsto 1 - \color{blue}{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)} \cdot \left(\left(1 - 0.3275911 \cdot \left|x\right|\right) \cdot e^{-x \cdot x}\right)} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 79.2%

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

Alternative 6: 79.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] - N[(1.453152027 / t$95$1), $MachinePrecision]), $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 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

6. Applied rewrites 79.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. associate-/r* N/A

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

8. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

6. Applied rewrites 79.2%

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

Alternative 8: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return fma(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) / fma(-0.3275911, abs(x), -1.0)), exp(Float64(Float64(-x) * x)), 1.0)
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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 = fma(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))));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - 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(Float64(x * x)))))
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 10: 78.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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)
     (+
      (fma (* x x) (fma 0.3275911 (fabs x) 1.0) (* 0.3275911 (fabs x)))
      1.0)))))

C

double code(double x) {
	double t_0 = fma(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) / (fma((x * x), fma(0.3275911, fabs(x), 1.0), (0.3275911 * fabs(x))) + 1.0));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - 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(fma(Float64(x * x), fma(0.3275911, abs(x), 1.0), Float64(0.3275911 * abs(x))) + 1.0)))
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-*.f64 78.5

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

6. Applied rewrites 78.5%

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

Alternative 11: 78.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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 (fma x x 1.0))))))

C

double code(double x) {
	double t_0 = fma(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 * fma(x, x, 1.0)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - 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 * fma(x, x, 1.0))))
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.5

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

6. Applied rewrites 78.5%

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

Alternative 12: 77.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (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 = fma(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 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - 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 * 1.0)))
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.5%

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

Alternative 13: 55.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (/ 0.254829592 (* (exp (* x x)) (fma 0.3275911 (fabs x) 1.0)))))

C

double code(double x) {
	return 1.0 - (0.254829592 / (exp((x * x)) * fma(0.3275911, fabs(x), 1.0)));
}

Julia

function code(x)
	return Float64(1.0 - Float64(0.254829592 / Float64(exp(Float64(x * x)) * fma(0.3275911, abs(x), 1.0))))
end

Wolfram

code[x_] := N[(1.0 - N[(0.254829592 / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{31853699}{125000000} \cdot \frac{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

6. Applied rewrites 55.6%

   \[\leadsto 1 - \color{blue}{\frac{0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]
7. Add Preprocessing

Alternative 14: 55.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.254829592}{\left(1 + x \cdot x\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (/ 0.254829592 (* (+ 1.0 (* x x)) (fma 0.3275911 (fabs x) 1.0)))))

C

double code(double x) {
	return 1.0 - (0.254829592 / ((1.0 + (x * x)) * fma(0.3275911, fabs(x), 1.0)));
}

Julia

function code(x)
	return Float64(1.0 - Float64(0.254829592 / Float64(Float64(1.0 + Float64(x * x)) * fma(0.3275911, abs(x), 1.0))))
end

Wolfram

code[x_] := N[(1.0 - N[(0.254829592 / N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{31853699}{125000000} \cdot \frac{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

6. Applied rewrites 55.6%

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

7. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{\left(1 + {x}^{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{3275911}{10000000}}, \left|x\right|, 1\right)} \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{\left(1 + {x}^{2}\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   2. pow2 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{\left(1 + x \cdot x\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   3. lift-*.f64 55.3

      \[\leadsto 1 - \frac{0.254829592}{\left(1 + x \cdot x\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

9. Applied rewrites 55.3%

   \[\leadsto 1 - \frac{0.254829592}{\left(1 + x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{0.3275911}, \left|x\right|, 1\right)} \]
10. Add Preprocessing

Alternative 15: 54.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.254829592}{1 - -0.3275911 \cdot \left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (/ 0.254829592 (- 1.0 (* -0.3275911 (fabs x))))))

C

double code(double x) {
	return 1.0 - (0.254829592 / (1.0 - (-0.3275911 * 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
    code = 1.0d0 - (0.254829592d0 / (1.0d0 - ((-0.3275911d0) * abs(x))))
end function

Java

public static double code(double x) {
	return 1.0 - (0.254829592 / (1.0 - (-0.3275911 * Math.abs(x))));
}

Python

def code(x):
	return 1.0 - (0.254829592 / (1.0 - (-0.3275911 * math.fabs(x))))

Julia

function code(x)
	return Float64(1.0 - Float64(0.254829592 / Float64(1.0 - Float64(-0.3275911 * abs(x)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (0.254829592 / (1.0 - (-0.3275911 * abs(x))));
end

Wolfram

code[x_] := N[(1.0 - N[(0.254829592 / N[(1.0 - N[(-0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{31853699}{125000000} \cdot \frac{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

6. Applied rewrites 55.6%

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

7. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{\color{blue}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{1 + \color{blue}{\frac{3275911}{10000000} \cdot \left|x\right|}} \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{1 - \left(\mathsf{neg}\left(\frac{3275911}{10000000}\right)\right) \cdot \color{blue}{\left|x\right|}} \]

   3. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|} \]

   4. lower--.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{1 - \frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|}} \]

   5. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|} \]

   6. lift-fabs.f64 54.5

      \[\leadsto 1 - \frac{0.254829592}{1 - -0.3275911 \cdot \left|x\right|} \]

9. Applied rewrites 54.5%

   \[\leadsto 1 - \frac{0.254829592}{\color{blue}{1 - -0.3275911 \cdot \left|x\right|}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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)))))))