Jmat.Real.erf

Percentage Accurate: 79.2% → 79.2%

Time: 9.8s

Alternatives: 14

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: 79.2% 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: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(\frac{-0.284496736}{t\_0} - \frac{-1.421413741 - \frac{1.453152027 - \frac{1.061405429}{t\_0}}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{0.10731592879921 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-0.6551822, \left|x\right|, -1\right)}\right) - -0.254829592\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (-
    1.0
    (*
     (*
      (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
      (-
       (-
        (/ -0.284496736 t_0)
        (/
         (-
          -1.421413741
          (/
           (- 1.453152027 (/ 1.061405429 t_0))
           (fma (fabs x) -0.3275911 -1.0)))
         (- (* 0.10731592879921 (* x x)) (fma -0.6551822 (fabs x) -1.0))))
       -0.254829592))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (((-0.284496736 / t_0) - ((-1.421413741 - ((1.453152027 - (1.061405429 / t_0)) / fma(fabs(x), -0.3275911, -1.0))) / ((0.10731592879921 * (x * x)) - fma(-0.6551822, fabs(x), -1.0)))) - -0.254829592)) * exp(-(fabs(x) * fabs(x))));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(Float64(Float64(-0.284496736 / t_0) - Float64(Float64(-1.421413741 - Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(abs(x), -0.3275911, -1.0))) / Float64(Float64(0.10731592879921 * Float64(x * x)) - fma(-0.6551822, abs(x), -1.0)))) - -0.254829592)) * 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]}, N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.284496736 / t$95$0), $MachinePrecision] - N[(N[(-1.421413741 - N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(-0.6551822 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fma.f64 N/A

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

   4. sum-square-pow N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.2%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - \frac{-1.421413741 - \frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{0.10731592879921 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-0.6551822, \left|x\right|, -1\right)}\right) - -0.254829592\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
8. Add Preprocessing

Alternative 2: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{e^{\left(-x\right) \cdot x}}{t\_0} \cdot \left(\left(\frac{-0.284496736}{t\_0} - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{t\_0}}{t\_0}}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, 1 - -0.6551822 \cdot \left|x\right|\right)}\right) - -0.254829592\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (/ (exp (* (- x) x)) t_0)
     (-
      (-
       (/ -0.284496736 t_0)
       (/
        (- -1.421413741 (/ (- -1.453152027 (/ -1.061405429 t_0)) t_0))
        (fma (* 0.10731592879921 x) x (- 1.0 (* -0.6551822 (fabs x))))))
      -0.254829592)))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((exp((-x * x)) / t_0) * (((-0.284496736 / t_0) - ((-1.421413741 - ((-1.453152027 - (-1.061405429 / t_0)) / t_0)) / fma((0.10731592879921 * x), x, (1.0 - (-0.6551822 * fabs(x)))))) - -0.254829592));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(exp(Float64(Float64(-x) * x)) / t_0) * Float64(Float64(Float64(-0.284496736 / t_0) - Float64(Float64(-1.421413741 - Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_0)) / t_0)) / fma(Float64(0.10731592879921 * x), x, Float64(1.0 - Float64(-0.6551822 * abs(x)))))) - -0.254829592)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(-0.284496736 / t$95$0), $MachinePrecision] - N[(N[(-1.421413741 - N[(N[(-1.453152027 - N[(-1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(1.0 - N[(-0.6551822 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fma.f64 N/A

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

   4. sum-square-pow N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.2%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - \frac{-1.421413741 - \frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{0.10731592879921 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-0.6551822, \left|x\right|, -1\right)}\right) - -0.254829592\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

9. Applied rewrites 79.2%

   \[\leadsto 1 - \color{blue}{\frac{e^{\left(-x\right) \cdot x}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot \left(\left(\frac{-0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, 1 - -0.6551822 \cdot \left|x\right|\right)}\right) - -0.254829592\right)} \]
10. Add Preprocessing

Alternative 3: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\frac{\left(\frac{-0.284496736}{t\_0} - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{t\_0}}{t\_0}}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, 1 - -0.6551822 \cdot \left|x\right|\right)}\right) - -0.254829592}{e^{x \cdot x}}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (/
     (/
      (-
       (-
        (/ -0.284496736 t_0)
        (/
         (- -1.421413741 (/ (- -1.453152027 (/ -1.061405429 t_0)) t_0))
         (fma (* 0.10731592879921 x) x (- 1.0 (* -0.6551822 (fabs x))))))
       -0.254829592)
      (exp (* x x)))
     t_0))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - (((((-0.284496736 / t_0) - ((-1.421413741 - ((-1.453152027 - (-1.061405429 / t_0)) / t_0)) / fma((0.10731592879921 * x), x, (1.0 - (-0.6551822 * fabs(x)))))) - -0.254829592) / exp((x * x))) / t_0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(-0.284496736 / t_0) - Float64(Float64(-1.421413741 - Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_0)) / t_0)) / fma(Float64(0.10731592879921 * x), x, Float64(1.0 - Float64(-0.6551822 * abs(x)))))) - -0.254829592) / exp(Float64(x * x))) / t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(-0.284496736 / t$95$0), $MachinePrecision] - N[(N[(-1.421413741 - N[(N[(-1.453152027 - N[(-1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(1.0 - N[(-0.6551822 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fma.f64 N/A

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

   4. sum-square-pow N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.2%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - \frac{-1.421413741 - \frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{0.10731592879921 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-0.6551822, \left|x\right|, -1\right)}\right) - -0.254829592\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

9. Applied rewrites 79.2%

   \[\leadsto 1 - \color{blue}{\frac{\frac{\left(\frac{-0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, 1 - -0.6551822 \cdot \left|x\right|\right)}\right) - -0.254829592}{e^{x \cdot x}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \]
10. Add Preprocessing

Alternative 4: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\left(\frac{-0.284496736}{t\_0} - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{t\_0}}{t\_0}}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, 1 - -0.6551822 \cdot \left|x\right|\right)}\right) - -0.254829592}{e^{x \cdot x} \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (-
       (/ -0.284496736 t_0)
       (/
        (- -1.421413741 (/ (- -1.453152027 (/ -1.061405429 t_0)) t_0))
        (fma (* 0.10731592879921 x) x (- 1.0 (* -0.6551822 (fabs x))))))
      -0.254829592)
     (* (exp (* x x)) t_0)))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((((-0.284496736 / t_0) - ((-1.421413741 - ((-1.453152027 - (-1.061405429 / t_0)) / t_0)) / fma((0.10731592879921 * x), x, (1.0 - (-0.6551822 * fabs(x)))))) - -0.254829592) / (exp((x * x)) * t_0));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(-0.284496736 / t_0) - Float64(Float64(-1.421413741 - Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_0)) / t_0)) / fma(Float64(0.10731592879921 * x), x, Float64(1.0 - Float64(-0.6551822 * abs(x)))))) - -0.254829592) / Float64(exp(Float64(x * x)) * t_0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(-0.284496736 / t$95$0), $MachinePrecision] - N[(N[(-1.421413741 - N[(N[(-1.453152027 - N[(-1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(1.0 - N[(-0.6551822 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fma.f64 N/A

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

   4. sum-square-pow N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.2%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - \frac{-1.421413741 - \frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{0.10731592879921 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-0.6551822, \left|x\right|, -1\right)}\right) - -0.254829592\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

9. Applied rewrites 79.2%

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{-0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, 1 - -0.6551822 \cdot \left|x\right|\right)}\right) - -0.254829592}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \]
10. Add Preprocessing

Alternative 5: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

   1. lift-exp.f64 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f64 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. exp-lft-sqr-rev N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-*.f64 79.2

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

7. Applied rewrites 79.2%

   \[\leadsto 1 - \frac{\frac{\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, -1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.284496736\right)}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -0.254829592}{\color{blue}{\sqrt{e^{\left(x \cdot x\right) \cdot 2}}} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-lft-sqr N/A

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

   4. lift-*.f64 N/A

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

   5. exp-prod N/A

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

   6. lift-*.f64 N/A

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

   7. exp-prod N/A

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

   8. pow-prod-up N/A

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

   9. pow-exp N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 79.2

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

9. Applied rewrites 79.2%

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

Alternative 6: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

Alternative 8: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

5. Applied rewrites 79.2%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 10: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = fma(-0.3275911, fabs(x), -1.0);
	return 1.0 - (((-0.254829592 - (fma((-1.0 / t_1), (-1.421413741 - ((-1.453152027 - (-1.061405429 / t_0)) / t_0)), 0.284496736) / t_1)) / t_1) * 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(-0.254829592 - Float64(fma(Float64(-1.0 / t_1), Float64(-1.421413741 - Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_0)) / t_0)), 0.284496736) / t_1)) / t_1) * 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[(-0.254829592 - N[(N[(N[(-1.0 / t$95$1), $MachinePrecision] * N[(-1.421413741 - N[(N[(-1.453152027 - N[(-1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + 0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.6%

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

Alternative 11: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.6%

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

10. Applied rewrites 77.6%

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

Alternative 12: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.6%

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

10. Applied rewrites 77.6%

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

Alternative 13: 77.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.6%

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

10. Applied rewrites 77.6%

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

Alternative 14: 55.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = 1.0 + (0.3275911 * fabs(x));
	return 1.0 - (((0.254829592 - (0.284496736 * (1.0 / t_0))) / t_0) * 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 + (0.3275911d0 * abs(x))
    code = 1.0d0 - (((0.254829592d0 - (0.284496736d0 * (1.0d0 / t_0))) / t_0) * exp(-(abs(x) * abs(x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 78.0%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fma.f64 N/A

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

   4. sum-square-pow N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.2%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - \frac{-1.421413741 - \frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{0.10731592879921 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(-0.6551822, \left|x\right|, -1\right)}\right) - -0.254829592\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

8. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{\color{blue}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. lower--.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{\color{blue}{1} + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. lower-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. lower-+.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. lower-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. lower-+.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \color{blue}{\frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   9. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \color{blue}{\left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   10. lower-fabs.f64 55.7

       \[\leadsto 1 - \frac{0.254829592 - 0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

10. Applied rewrites 55.7%

    \[\leadsto 1 - \color{blue}{\frac{0.254829592 - 0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))))))