Jmat.Real.erf

Percentage Accurate: 79.3% → 80.5%

Time: 10.6s

Alternatives: 15

Speedup: 1.2×

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: 79.3% 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: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := {t\_1}^{-1}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), t\_2, 0.254829592\right)\\ t_4 := \left(t\_0 \cdot t\_3\right) \cdot t\_2\\ t_5 := 1 + \left({t\_4}^{6} + {t\_4}^{3}\right)\\ t_6 := t\_0 \cdot \left(t\_3 \cdot t\_2\right)\\ \frac{\frac{1}{t\_5} - \frac{{t\_4}^{9}}{t\_5}}{1 + \left({t\_6}^{2} + t\_6\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (pow t_1 -1.0))
        (t_3
         (fma
          (fma
           (- 1.0 (* (fabs x) 0.3275911))
           (/
            (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
            (fma (* x x) -0.10731592879921 1.0))
           -0.284496736)
          t_2
          0.254829592))
        (t_4 (* (* t_0 t_3) t_2))
        (t_5 (+ 1.0 (+ (pow t_4 6.0) (pow t_4 3.0))))
        (t_6 (* t_0 (* t_3 t_2))))
   (/ (- (/ 1.0 t_5) (/ (pow t_4 9.0) t_5)) (+ 1.0 (+ (pow t_6 2.0) t_6)))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = pow(t_1, -1.0);
	double t_3 = fma(fma((1.0 - (fabs(x) * 0.3275911)), (((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / fma((x * x), -0.10731592879921, 1.0)), -0.284496736), t_2, 0.254829592);
	double t_4 = (t_0 * t_3) * t_2;
	double t_5 = 1.0 + (pow(t_4, 6.0) + pow(t_4, 3.0));
	double t_6 = t_0 * (t_3 * t_2);
	return ((1.0 / t_5) - (pow(t_4, 9.0) / t_5)) / (1.0 + (pow(t_6, 2.0) + t_6));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = t_1 ^ -1.0
	t_3 = fma(fma(Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / fma(Float64(x * x), -0.10731592879921, 1.0)), -0.284496736), t_2, 0.254829592)
	t_4 = Float64(Float64(t_0 * t_3) * t_2)
	t_5 = Float64(1.0 + Float64((t_4 ^ 6.0) + (t_4 ^ 3.0)))
	t_6 = Float64(t_0 * Float64(t_3 * t_2))
	return Float64(Float64(Float64(1.0 / t_5) - Float64((t_4 ^ 9.0) / t_5)) / Float64(1.0 + Float64((t_6 ^ 2.0) + t_6)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] * t$95$2 + 0.254829592), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 + N[(N[Power[t$95$4, 6.0], $MachinePrecision] + N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$0 * N[(t$95$3 * t$95$2), $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[(1.0 + N[(N[Power[t$95$6, 2.0], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\frac{1 - {\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)}} \]

8. Applied rewrites 79.7%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}^{3}}{1 + \mathsf{fma}\left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}, {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}, 1 \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)} \]

10. Applied rewrites 80.8%

    \[\leadsto \frac{\color{blue}{\frac{1}{1 + \left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{6} + 1 \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)} - \frac{{\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{9}}{1 + \left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{6} + 1 \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)} \]

11. Final simplification 80.8%

    \[\leadsto \frac{\frac{1}{1 + \left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{6} + {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)} - \frac{{\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{9}}{1 + \left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{6} + {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)} \]
12. Add Preprocessing

Alternative 2: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := {t\_1}^{-1}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), t\_2, 0.254829592\right)\\ t_4 := {\left(\left(t\_0 \cdot t\_3\right) \cdot t\_2\right)}^{3}\\ t_5 := t\_0 \cdot \left(t\_3 \cdot t\_2\right)\\ \frac{\frac{1 - t\_4 \cdot t\_4}{1 + t\_4}}{1 + \left({t\_5}^{2} + t\_5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (pow t_1 -1.0))
        (t_3
         (fma
          (fma
           (- 1.0 (* (fabs x) 0.3275911))
           (/
            (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
            (fma (* x x) -0.10731592879921 1.0))
           -0.284496736)
          t_2
          0.254829592))
        (t_4 (pow (* (* t_0 t_3) t_2) 3.0))
        (t_5 (* t_0 (* t_3 t_2))))
   (/ (/ (- 1.0 (* t_4 t_4)) (+ 1.0 t_4)) (+ 1.0 (+ (pow t_5 2.0) t_5)))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = pow(t_1, -1.0);
	double t_3 = fma(fma((1.0 - (fabs(x) * 0.3275911)), (((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / fma((x * x), -0.10731592879921, 1.0)), -0.284496736), t_2, 0.254829592);
	double t_4 = pow(((t_0 * t_3) * t_2), 3.0);
	double t_5 = t_0 * (t_3 * t_2);
	return ((1.0 - (t_4 * t_4)) / (1.0 + t_4)) / (1.0 + (pow(t_5, 2.0) + t_5));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = t_1 ^ -1.0
	t_3 = fma(fma(Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / fma(Float64(x * x), -0.10731592879921, 1.0)), -0.284496736), t_2, 0.254829592)
	t_4 = Float64(Float64(t_0 * t_3) * t_2) ^ 3.0
	t_5 = Float64(t_0 * Float64(t_3 * t_2))
	return Float64(Float64(Float64(1.0 - Float64(t_4 * t_4)) / Float64(1.0 + t_4)) / Float64(1.0 + Float64((t_5 ^ 2.0) + t_5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] * t$95$2 + 0.254829592), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[(t$95$0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$5, 2.0], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\frac{1 - {\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)}} \]

8. Applied rewrites 79.7%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3} \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}}{1 + {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}}}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)} \]

9. Final simplification 79.7%

   \[\leadsto \frac{\frac{1 - {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3} \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}}{1 + {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)} \]
10. Add Preprocessing

Alternative 3: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := {t\_1}^{-1}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), t\_2, 0.254829592\right)\\ t_4 := \left(t\_0 \cdot t\_3\right) \cdot t\_2\\ t_5 := t\_0 \cdot \left(t\_3 \cdot t\_2\right)\\ \frac{\frac{1 - {t\_4}^{9}}{1 + \left({t\_4}^{6} + {t\_4}^{3}\right)}}{1 + \left({t\_5}^{2} + t\_5\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (pow t_1 -1.0))
        (t_3
         (fma
          (fma
           (- 1.0 (* (fabs x) 0.3275911))
           (/
            (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
            (fma (* x x) -0.10731592879921 1.0))
           -0.284496736)
          t_2
          0.254829592))
        (t_4 (* (* t_0 t_3) t_2))
        (t_5 (* t_0 (* t_3 t_2))))
   (/
    (/ (- 1.0 (pow t_4 9.0)) (+ 1.0 (+ (pow t_4 6.0) (pow t_4 3.0))))
    (+ 1.0 (+ (pow t_5 2.0) t_5)))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = pow(t_1, -1.0);
	double t_3 = fma(fma((1.0 - (fabs(x) * 0.3275911)), (((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / fma((x * x), -0.10731592879921, 1.0)), -0.284496736), t_2, 0.254829592);
	double t_4 = (t_0 * t_3) * t_2;
	double t_5 = t_0 * (t_3 * t_2);
	return ((1.0 - pow(t_4, 9.0)) / (1.0 + (pow(t_4, 6.0) + pow(t_4, 3.0)))) / (1.0 + (pow(t_5, 2.0) + t_5));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = t_1 ^ -1.0
	t_3 = fma(fma(Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / fma(Float64(x * x), -0.10731592879921, 1.0)), -0.284496736), t_2, 0.254829592)
	t_4 = Float64(Float64(t_0 * t_3) * t_2)
	t_5 = Float64(t_0 * Float64(t_3 * t_2))
	return Float64(Float64(Float64(1.0 - (t_4 ^ 9.0)) / Float64(1.0 + Float64((t_4 ^ 6.0) + (t_4 ^ 3.0)))) / Float64(1.0 + Float64((t_5 ^ 2.0) + t_5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] * t$95$2 + 0.254829592), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * 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[(N[Power[t$95$4, 6.0], $MachinePrecision] + N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$5, 2.0], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\frac{1 - {\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)}} \]

8. Applied rewrites 79.7%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}^{3}}{1 + \mathsf{fma}\left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}, {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}, 1 \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)} \]

10. Applied rewrites 79.7%

    \[\leadsto \color{blue}{\frac{\frac{1 - {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{9}}{1 + \left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{6} + 1 \cdot {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + 1 \cdot \left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)\right)}} \]

11. Final simplification 79.7%

    \[\leadsto \frac{\frac{1 - {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{9}}{1 + \left({\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{6} + {\left(\left(e^{\left(-x\right) \cdot x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right)\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}^{3}\right)}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2} + e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \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(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)} \]
12. Add Preprocessing

Alternative 4: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := {t\_0}^{-1}\\ t_2 := e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), t\_1, 0.254829592\right) \cdot t\_1\right)\\ t_3 := 1 + t\_2\\ \frac{1}{t\_3} - \frac{{t\_2}^{2}}{t\_3} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (pow t_0 -1.0))
        (t_2
         (*
          (exp (* (- x) x))
          (*
           (fma
            (fma
             (- 1.0 (* (fabs x) 0.3275911))
             (/
              (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
              (fma (* x x) -0.10731592879921 1.0))
             -0.284496736)
            t_1
            0.254829592)
           t_1)))
        (t_3 (+ 1.0 t_2)))
   (- (/ 1.0 t_3) (/ (pow t_2 2.0) t_3))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = pow(t_0, -1.0);
	double t_2 = exp((-x * x)) * (fma(fma((1.0 - (fabs(x) * 0.3275911)), (((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma((x * x), -0.10731592879921, 1.0)), -0.284496736), t_1, 0.254829592) * t_1);
	double t_3 = 1.0 + t_2;
	return (1.0 / t_3) - (pow(t_2, 2.0) / t_3);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = t_0 ^ -1.0
	t_2 = Float64(exp(Float64(Float64(-x) * x)) * Float64(fma(fma(Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma(Float64(x * x), -0.10731592879921, 1.0)), -0.284496736), t_1, 0.254829592) * t_1))
	t_3 = Float64(1.0 + t_2)
	return Float64(Float64(1.0 / t_3) - Float64((t_2 ^ 2.0) / t_3))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, -1.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] * t$95$1 + 0.254829592), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, N[(N[(1.0 / t$95$3), $MachinePrecision] - N[(N[Power[t$95$2, 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\frac{1 - {\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2}}{1 + e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}} \]

8. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)} - \frac{{\left(e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)\right)}^{2}}{1 + e^{\left(-x\right) \cdot x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, -0.284496736\right), {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.254829592\right) \cdot {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}\right)}} \]
9. Add Preprocessing

Alternative 5: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + ((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / (1.0 - (0.10731592879921 * (x * x)))) * (1.0 - (fabs(x) * 0.3275911))))))) * (1.0 / exp((x * x))));
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))) * Float64(1.0 - Float64(abs(x) * 0.3275911))))))) * Float64(1.0 / exp(Float64(x * 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]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. pow2 N/A

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

   10. lower-exp.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 79.6

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

6. Applied rewrites 79.6%

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

Alternative 6: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + ((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / (1.0 - (0.10731592879921 * (x * x)))) * (1.0 - (fabs(x) * 0.3275911))))))) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))) * Float64(1.0 - Float64(abs(x) * 0.3275911))))))) * exp(Float64(Float64(-x) * 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]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

5. Final simplification 79.6%

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

Alternative 7: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = 1.0 + (0.3275911 * fabs(x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 + (((-1.0 / t_0) * (0.254829592 + ((1.0 / t_0) * fma((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / (1.0 - (0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), -0.284496736)))) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 + Float64(Float64(Float64(-1.0 / t_0) * Float64(0.254829592 + Float64(Float64(1.0 / t_0) * fma(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), -0.284496736)))) * exp(Float64(Float64(-x) * x))))
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[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 + N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

5. Final simplification 79.6%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

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

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) -1.421413741) t_0)
         -0.284496736)
        t_0)
       0.254829592)
      (fma (* x x) -0.10731592879921 1.0))
     (* (- 1.0 (* (fabs x) 0.3275911)) (/ 1.0 (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) / fma((x * x), -0.10731592879921, 1.0)) * ((1.0 - (fabs(x) * 0.3275911)) * (1.0 / 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(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / fma(Float64(x * x), -0.10731592879921, 1.0)) * Float64(Float64(1.0 - Float64(abs(x) * 0.3275911)) * Float64(1.0 / 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[(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[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. sqr-abs-rev N/A

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

   8. pow2 N/A

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

   9. exp-neg N/A

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

   10. lower-/.f64 N/A

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

   11. pow2 N/A

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

   12. sqr-abs-rev N/A

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

   13. pow2 N/A

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

   14. lower-exp.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 79.6

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

8. Applied rewrites 79.6%

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

Alternative 9: 79.3% accurate, 1.1× speedup

Math

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

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) -1.421413741) t_0)
         -0.284496736)
        t_0)
       0.254829592)
      (fma (* x x) -0.10731592879921 1.0))
     (* (- 1.0 (* (fabs x) 0.3275911)) (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) / fma((x * x), -0.10731592879921, 1.0)) * ((1.0 - (fabs(x) * 0.3275911)) * 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) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / fma(Float64(x * x), -0.10731592879921, 1.0)) * Float64(Float64(1.0 - Float64(abs(x) * 0.3275911)) * 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] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

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

Alternative 10: 79.3% accurate, 1.2× speedup

Math

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

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(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp(Float64(Float64(-x) * x))))
end

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] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

Alternative 11: 79.3% accurate, 1.2× 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 79.6%

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

4. Applied rewrites 79.6%

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

Alternative 12: 77.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fabs(x) * 0.3275911;
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((((((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / (((t_0 * t_0) - 1.0) / (t_0 - 1.0))) + -0.284496736) / t_1) + 0.254829592) / fma((x * x), -0.10731592879921, 1.0)) * (1.0 - (0.3275911 * fabs(x))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-fabs.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. lift-*.f64 79.6

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

8. Applied rewrites 79.6%

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

9. Taylor expanded in x around 0

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

    1. lower--.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lift-fabs.f64 78.2

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

11. Applied rewrites 78.2%

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

Alternative 13: 77.7% accurate, 2.0× speedup

Math

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

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)
      (fma (* x x) -0.10731592879921 1.0))
     (- 1.0 (* 0.3275911 (fabs 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) / fma((x * x), -0.10731592879921, 1.0)) * (1.0 - (0.3275911 * fabs(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) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / fma(Float64(x * x), -0.10731592879921, 1.0)) * Float64(1.0 - Float64(0.3275911 * abs(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] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

6. Applied rewrites 79.6%

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

7. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-fabs.f64 78.2

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

9. Applied rewrites 78.2%

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

Alternative 14: 77.7% accurate, 2.3× speedup

Math

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

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))
    1.0
    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)), 1.0, 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)), 1.0, 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] * 1.0 + 1.0), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

5. Taylor expanded in x around 0

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

   1. distribute-lft-neg-out 78.2

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

   2. pow2 78.2

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

   3. pow2 78.2

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

   4. sqr-abs-rev 78.2

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

7. Applied rewrites 78.2%

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

Alternative 15: 55.7% accurate, 262.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 79.6%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 56.3%

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

Reproduce

herbie shell --seed 2025085 
(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)))))))