Jmat.Real.erf

Percentage Accurate: 79.3% → 80.4%

Time: 8.0s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	t_1 = fma(-0.3275911, abs(x), -1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(1.0 / Float64(t_1 / Float64(Float64(Float64(Float64(Float64(-1.453152027 - Float64(1.061405429 / t_1)) / t_1) - 1.421413741) / t_3) - -0.284496736))) - -0.254829592) / Float64(t_0 * t_3))
	return Float64(Float64((1.0 ^ 3.0) - Float64(1.0 / Float64((Float64(t_0 * t_2) ^ 3.0) / (Float64(Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_2) - -1.453152027) / t_1) - -1.421413741) / t_1)) / t_2) - -0.254829592) ^ 3.0)))) / fma(1.0, 1.0, fma(t_4, t_4, t_4)))
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]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(1.0 / N[(t$95$1 / N[(N[(N[(N[(N[(-1.453152027 - N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[(1.0 / N[(N[Power[N[(t$95$0 * t$95$2), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(N[(N[(-0.284496736 - N[(N[(N[(N[(N[(-1.061405429 / t$95$2), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -0.254829592), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 * 1.0 + N[(t$95$4 * t$95$4 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 80.4%

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

7. Applied rewrites 80.4%

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

9. Applied rewrites 80.4%

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

11. Applied rewrites 80.4%

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

Alternative 2: 80.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = fma(-0.3275911, abs(x), -1.0)
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_3) - -1.453152027) / t_3) - 1.421413741) / t_3) - -0.284496736) / t_2) - -0.254829592) / Float64(t_0 * t_3))
	return Float64(Float64((1.0 ^ 3.0) - Float64(1.0 / Float64((Float64(t_0 * t_1) ^ 3.0) / (Float64(Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_1) - -1.453152027) / t_2) - -1.421413741) / t_2)) / t_1) - -0.254829592) ^ 3.0)))) / fma(1.0, 1.0, fma(t_4, t_4, t_4)))
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]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / t$95$3), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$3), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[(1.0 / N[(N[Power[N[(t$95$0 * t$95$1), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(N[(N[(-0.284496736 - N[(N[(N[(N[(N[(-1.061405429 / t$95$1), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - -0.254829592), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 * 1.0 + N[(t$95$4 * t$95$4 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 80.4%

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

Alternative 3: 80.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	t_1 = fma(-0.3275911, abs(x), -1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.453152027 - Float64(1.061405429 / t_1)) / t_1) - 1.421413741) / t_3) - -0.284496736) / t_1) - -0.254829592) / Float64(t_3 * t_0))
	return Float64(Float64((1.0 ^ 3.0) - Float64(1.0 / Float64((Float64(t_0 * t_2) ^ 3.0) / (Float64(Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_2) - -1.453152027) / t_1) - -1.421413741) / t_1)) / t_2) - -0.254829592) ^ 3.0)))) / fma(Float64(t_4 + 1.0), t_4, 1.0))
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]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(-1.453152027 - N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[(1.0 / N[(N[Power[N[(t$95$0 * t$95$2), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(N[(N[(-0.284496736 - N[(N[(N[(N[(N[(-1.061405429 / t$95$2), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -0.254829592), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 + 1.0), $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 80.4%

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

7. Applied rewrites 80.4%

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

Alternative 4: 79.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 5: 79.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 79.3%

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

Alternative 6: 79.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

Alternative 7: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

Alternative 8: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 9: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 10: 77.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.6%

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

8. Applied rewrites 77.6%

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

Alternative 11: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(Float64(Float64(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) / Float64(fma(abs(x), 0.3275911, 1.0) * 1.0)))
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.6%

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

8. Applied rewrites 77.6%

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

Alternative 12: 77.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (-
        (/ (- (/ (- (/ -1.061405429 t_0) -1.453152027) t_0) 1.421413741) t_0)
        -0.284496736)
       (fma -0.3275911 (fabs x) -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) - -1.453152027) / t_0) - 1.421413741) / t_0) - -0.284496736) / fma(-0.3275911, fabs(x), -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(Float64(-1.061405429 / t_0) - -1.453152027) / t_0) - 1.421413741) / t_0) - -0.284496736) / fma(-0.3275911, abs(x), -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[(N[(-1.061405429 / t$95$0), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.284496736), $MachinePrecision] / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.6%

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

Reproduce

herbie shell --seed 2025166 
(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)))))))