Jmat.Real.erf

Percentage Accurate: 79.1% → 79.1%

Time: 9.0s

Alternatives: 8

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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.1%

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

Alternative 2: 79.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-fraction N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 79.1%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. mul-1-neg N/A

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

   5. lift-fma.f64 N/A

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

   6. add-flip N/A

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

   7. metadata-eval N/A

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

   8. sub-negate N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 79.1

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

   21. lift-fma.f64 N/A

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

7. Applied rewrites 79.1%

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

Alternative 3: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\mathsf{fma}\left(0.0834799063558312, \left|x\right|, 0.254829592\right) - \left(0.284496736 - \frac{-1.421413741 - \frac{\frac{1.061405429}{t\_0} - -1.453152027}{t\_0}}{t\_0}\right)}{t\_1 \cdot \left(e^{x \cdot x} \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (fma 0.0834799063558312 (fabs x) 0.254829592)
      (-
       0.284496736
       (/ (- -1.421413741 (/ (- (/ 1.061405429 t_0) -1.453152027) t_0)) t_0)))
     (* t_1 (* (exp (* x x)) t_1))))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((fma(0.0834799063558312, fabs(x), 0.254829592) - (0.284496736 - ((-1.421413741 - (((1.061405429 / t_0) - -1.453152027) / t_0)) / t_0))) / (t_1 * (exp((x * x)) * t_1)));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(fma(0.0834799063558312, abs(x), 0.254829592) - Float64(0.284496736 - Float64(Float64(-1.421413741 - Float64(Float64(Float64(1.061405429 / t_0) - -1.453152027) / t_0)) / t_0))) / Float64(t_1 * Float64(exp(Float64(x * x)) * t_1))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(0.0834799063558312 * N[Abs[x], $MachinePrecision] + 0.254829592), $MachinePrecision] - N[(0.284496736 - N[(N[(-1.421413741 - N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 77.9%

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

5. Applied rewrites 79.1%

   \[\leadsto 1 - \frac{\color{blue}{\mathsf{fma}\left(0.0834799063558312, \left|x\right|, 0.254829592\right) - \left(0.284496736 - \frac{-1.421413741 - \frac{\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \left(e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)} \]
6. Add Preprocessing

Alternative 4: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

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

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

   7. lift--.f64 N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 79.1%

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

Alternative 5: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

Alternative 6: 77.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. sub-negate N/A

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

   17. metadata-eval N/A

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

   18. add-flip N/A

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

   19. lift-fma.f64 N/A

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

8. Applied rewrites 77.3%

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

Alternative 7: 77.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.3%

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

8. Applied rewrites 77.3%

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

Alternative 8: 77.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.3%

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

Reproduce

herbie shell --seed 2025147 
(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)))))))