Jmat.Real.erf

Percentage Accurate: 79.5% → 79.5%

Time: 9.6s

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: 79.5% 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: 79.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

5. Applied rewrites 79.5%

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

7. Applied rewrites 79.5%

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

9. Applied rewrites 79.5%

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

Alternative 2: 79.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

5. Applied rewrites 79.5%

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

Alternative 3: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

Alternative 4: 77.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.9%

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

8. Applied rewrites 77.9%

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

Alternative 5: 77.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.9%

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

8. Applied rewrites 77.9%

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

Alternative 6: 77.9% accurate, 1.5× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.9%

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

8. Applied rewrites 77.9%

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

Alternative 7: 77.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.9%

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

8. Applied rewrites 77.9%

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

Alternative 8: 77.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.5%

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.9%

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

8. Applied rewrites 77.9%

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

Reproduce

herbie shell --seed 2025161 
(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)))))))