Jmat.Real.erf

Percentage Accurate: 78.4% → 78.4%

Time: 7.1s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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: 78.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Applied rewrites 78.4%

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

5. Applied rewrites 78.4%

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

7. Applied rewrites 78.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. sqr-abs-rev N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 78.4

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

9. Applied rewrites 78.4%

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

Alternative 2: 78.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Applied rewrites 78.4%

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

5. Applied rewrites 78.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs-rev N/A

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

   5. lift-*.f64 78.4

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

7. Applied rewrites 78.4%

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

Alternative 3: 78.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Applied rewrites 78.4%

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

   1. lift-/.f64 N/A

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

   2. division-flip N/A

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

   3. lower-special-/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-special-/ N/A

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

   10. lower-/.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fabs.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-fma.f64 78.4

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

5. Applied rewrites 78.4%

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

Alternative 4: 78.4% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.4%

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

3. Applied rewrites 78.4%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fabs.f64 N/A

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

   4. pow2-fabs-rev N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 78.4

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

5. Applied rewrites 78.4%

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

7. Applied rewrites 78.4%

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

Alternative 5: 78.4% 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(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 78.4%

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

3. Applied rewrites 78.4%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-fabs.f64 N/A

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

   4. pow2-fabs-rev N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 78.4

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

5. Applied rewrites 78.4%

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

Alternative 6: 78.4% 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 78.4%

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

3. Applied rewrites 78.4%

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

Alternative 7: 76.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), -0.3275911, -1.0)
	t_1 = fma(0.3275911, abs(x), 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)) / fma(abs(x), 0.3275911, 1.0)) - -0.254829592) / fma(-0.3275911, abs(x), -1.0)), 1.0, 1.0)
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Applied rewrites 29.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 76.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 76.9%

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

Alternative 8: 76.9% accurate, 1.6× speedup

Math

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

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)
    1.0
    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), 1.0, 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), 1.0, 1.0)
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Applied rewrites 29.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 76.9%

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

Reproduce

herbie shell --seed 2025132 
(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)))))))