Jmat.Real.erf

Percentage Accurate: 78.9% → 78.9%

Time: 6.6s

Alternatives: 13

Speedup: N/A×

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: 78.9% 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: 78.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

Alternative 2: 78.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 78.9%

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

Alternative 3: 78.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

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

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

   10. metadata-eval N/A

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

   11. sub-negate-rev N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

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

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

   15. +-commutative N/A

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

5. Applied rewrites 78.9%

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

Alternative 4: 78.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

Alternative 5: 78.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

Alternative 6: 78.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

Alternative 7: 78.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 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)) / fma(-0.3275911, abs(x), -1.0))) / t_0)) / t_0) - -0.254829592) / Float64(fma(Float64(x * x), t_0, Float64(abs(x) * 0.3275911)) - -1.0)))
end

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 78.3

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

6. Applied rewrites 78.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. add-flip N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

8. Applied rewrites 78.3%

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

Alternative 8: 78.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 78.3

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

6. Applied rewrites 78.3%

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

8. Applied rewrites 78.3%

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

Alternative 9: 78.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.3

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

6. Applied rewrites 78.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

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

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

   10. metadata-eval N/A

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

   11. sub-negate-rev N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

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

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

   15. +-commutative N/A

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

8. Applied rewrites 78.3%

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

Alternative 10: 78.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.3

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

6. Applied rewrites 78.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 78.3%

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

Alternative 11: 78.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 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)) / fma(-0.3275911, abs(x), -1.0))) / t_0)) / t_0) - -0.254829592) / Float64(t_0 * fma(x, x, 1.0))))
end

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.3

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

6. Applied rewrites 78.3%

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

Alternative 12: 77.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 78.3

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

6. Applied rewrites 78.3%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-*.f64 77.2

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

9. Applied rewrites 77.2%

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

    1. lift--.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. lift-fabs.f64 N/A

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

    6. lift-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. +-commutative N/A

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

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

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

    10. metadata-eval N/A

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

    11. sub-negate-rev N/A

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

    12. mult-flip N/A

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

    13. metadata-eval N/A

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

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

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

    15. +-commutative N/A

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

11. Applied rewrites 77.2%

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

Alternative 13: 77.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 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)) / fma(abs(x), -0.3275911, -1.0))) / t_0)) / t_0) - -0.254829592) / Float64(Float64(abs(x) * 0.3275911) - -1.0)))
end

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Applied rewrites 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 78.3

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

6. Applied rewrites 78.3%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-*.f64 77.2

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

9. Applied rewrites 77.2%

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

11. Applied rewrites 77.2%

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

Reproduce

herbie shell --seed 2025134 
(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)))))))