Jmat.Real.erf

Percentage Accurate: 79.0% → 84.6%

Time: 16.4s

Alternatives: 10

Speedup: 0.9×

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: 79.0% 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: 84.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.1%

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

7. Applied rewrites 83.3%

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

9. Applied rewrites 84.6%

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

Alternative 2: 83.3% accurate, 0.4× speedup

Math

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

FPCore

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

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

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_1)) / fma(abs(x), -0.3275911, -1.0)) - -1.421413741) / Float64((t_1 ^ 3.0) * exp(Float64(x * x))))
	return Float64(Float64(Float64(1.0 - Float64(t_2 * t_2)) / Float64(1.0 + t_2)) - Float64(Float64(Float64(Float64(-0.284496736 / t_0) - -0.254829592) / t_0) * exp(Float64(Float64(-x) * x))))
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]}, Block[{t$95$2 = N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[Power[t$95$1, 3.0], $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-0.284496736 / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.1%

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

7. Applied rewrites 83.3%

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

Alternative 3: 82.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.1%

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

7. Applied rewrites 83.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 82.5%

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

Alternative 4: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \left(1 - \frac{\frac{-1.453152027 - \frac{-1.061405429}{t\_2}}{t\_2} - -1.421413741}{{t\_2}^{3}} \cdot t\_0\right) - \frac{\frac{t\_1 \cdot t\_1 - 0.06493812095888646}{t\_1 + -0.254829592}}{t\_2} \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (/ 0.284496736 (fma (fabs x) -0.3275911 -1.0)))
        (t_2 (fma 0.3275911 (fabs x) 1.0)))
   (-
    (-
     1.0
     (*
      (/
       (- (/ (- -1.453152027 (/ -1.061405429 t_2)) t_2) -1.421413741)
       (pow t_2 3.0))
      t_0))
    (*
     (/ (/ (- (* t_1 t_1) 0.06493812095888646) (+ t_1 -0.254829592)) t_2)
     t_0))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = 0.284496736 / fma(fabs(x), -0.3275911, -1.0);
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	return (1.0 - (((((-1.453152027 - (-1.061405429 / t_2)) / t_2) - -1.421413741) / pow(t_2, 3.0)) * t_0)) - (((((t_1 * t_1) - 0.06493812095888646) / (t_1 + -0.254829592)) / t_2) * t_0);
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = Float64(0.284496736 / fma(abs(x), -0.3275911, -1.0))
	t_2 = fma(0.3275911, abs(x), 1.0)
	return Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_2)) / t_2) - -1.421413741) / (t_2 ^ 3.0)) * t_0)) - Float64(Float64(Float64(Float64(Float64(t_1 * t_1) - 0.06493812095888646) / Float64(t_1 + -0.254829592)) / t_2) * t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.284496736 / N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(1.0 - N[(N[(N[(N[(N[(-1.453152027 - N[(-1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 0.06493812095888646), $MachinePrecision] / N[(t$95$1 + -0.254829592), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.1%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-frac-neg N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

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

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

   11. flip-- N/A

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

   12. lower-/.f64 N/A

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

7. Applied rewrites 79.6%

   \[\leadsto \left(1 - \frac{\frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} \cdot e^{\left(-x\right) \cdot x}\right) - \frac{\color{blue}{\frac{\frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} \cdot \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} - 0.06493812095888646}{\frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + -0.254829592}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x} \]
8. Add Preprocessing

Alternative 5: 79.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lift-neg.f64 N/A

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

   6. pow-flip N/A

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

   7. exp-prod N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 79.1

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

7. Applied rewrites 79.1%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lift-neg.f64 N/A

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

   6. pow-flip N/A

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

   7. exp-prod N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 79.1

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

9. Applied rewrites 79.1%

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

Alternative 6: 79.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.1%

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

Alternative 7: 79.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

Alternative 8: 78.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.4

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

6. Applied rewrites 78.4%

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

Alternative 9: 77.4% accurate, 1.6× speedup

Math

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

FPCore

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

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 fma((1.0 / t_1), (((((((1.453152027 - (1.061405429 / t_0)) / t_1) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592), 1.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 29.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.4%

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

Alternative 10: 77.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lift-fabs.f64 N/A

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

   3. lower-fma.f64 77.4

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

6. Applied rewrites 77.4%

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

Reproduce

herbie shell --seed 2025130 
(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)))))))