Jmat.Real.erf

Percentage Accurate: 79.0% → 80.2%

Time: 9.3s

Alternatives: 13

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_4 := t\_3 \cdot t\_0\\ t_5 := \frac{-1.061405429}{t\_1}\\ t_6 := -0.254829592 - \frac{-0.284496736 - \frac{\frac{t\_5 - -1.453152027}{t\_1} - 1.421413741}{t\_1}}{t\_1}\\ \frac{{1}^{3} - \frac{1}{\frac{{\left(t\_0 \cdot t\_2\right)}^{3}}{{\left(\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_2}}{t\_2} - 1.421413741}{t\_2} - -0.284496736}{t\_3} - -0.254829592\right)}^{3}}}}{\mathsf{fma}\left(\frac{{\left(\frac{t\_6}{t\_1 \cdot t\_0}\right)}^{2} - 1}{\frac{t\_6}{t\_4} - 1}, \frac{-0.254829592 - \frac{-0.284496736 - \frac{\frac{-1.453152027 - t\_5}{t\_3} - 1.421413741}{t\_1}}{t\_1}}{t\_4}, 1\right)} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (exp (* x x)))
       (t_1 (fma (fabs x) 0.3275911 1.0))
       (t_2 (fma 0.3275911 (fabs x) 1.0))
       (t_3 (fma -0.3275911 (fabs x) -1.0))
       (t_4 (* t_3 t_0))
       (t_5 (/ -1.061405429 t_1))
       (t_6
        (-
         -0.254829592
         (/
          (-
           -0.284496736
           (/ (- (/ (- t_5 -1.453152027) t_1) 1.421413741) t_1))
          t_1))))
  (/
   (-
    (pow 1.0 3.0)
    (/
     1.0
     (/
      (pow (* t_0 t_2) 3.0)
      (pow
       (-
        (/
         (-
          (/
           (- (/ (- 1.453152027 (/ 1.061405429 t_2)) t_2) 1.421413741)
           t_2)
          -0.284496736)
         t_3)
        -0.254829592)
       3.0))))
   (fma
    (/ (- (pow (/ t_6 (* t_1 t_0)) 2.0) 1.0) (- (/ t_6 t_4) 1.0))
    (/
     (-
      -0.254829592
      (/
       (-
        -0.284496736
        (/ (- (/ (- -1.453152027 t_5) t_3) 1.421413741) t_1))
       t_1))
     t_4)
    1.0))))

C

double code(double x) {
	double t_0 = exp((x * x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = t_3 * t_0;
	double t_5 = -1.061405429 / t_1;
	double t_6 = -0.254829592 - ((-0.284496736 - ((((t_5 - -1.453152027) / t_1) - 1.421413741) / t_1)) / t_1);
	return (pow(1.0, 3.0) - (1.0 / (pow((t_0 * t_2), 3.0) / pow((((((((1.453152027 - (1.061405429 / t_2)) / t_2) - 1.421413741) / t_2) - -0.284496736) / t_3) - -0.254829592), 3.0)))) / fma(((pow((t_6 / (t_1 * t_0)), 2.0) - 1.0) / ((t_6 / t_4) - 1.0)), ((-0.254829592 - ((-0.284496736 - ((((-1.453152027 - t_5) / t_3) - 1.421413741) / t_1)) / t_1)) / t_4), 1.0);
}

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(t_3 * t_0)
	t_5 = Float64(-1.061405429 / t_1)
	t_6 = Float64(-0.254829592 - Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(t_5 - -1.453152027) / t_1) - 1.421413741) / t_1)) / t_1))
	return Float64(Float64((1.0 ^ 3.0) - Float64(1.0 / Float64((Float64(t_0 * t_2) ^ 3.0) / (Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_2)) / t_2) - 1.421413741) / t_2) - -0.284496736) / t_3) - -0.254829592) ^ 3.0)))) / fma(Float64(Float64((Float64(t_6 / Float64(t_1 * t_0)) ^ 2.0) - 1.0) / Float64(Float64(t_6 / t_4) - 1.0)), Float64(Float64(-0.254829592 - Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(-1.453152027 - t_5) / t_3) - 1.421413741) / t_1)) / t_1)) / t_4), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(-1.061405429 / t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(-0.254829592 - N[(N[(-0.284496736 - N[(N[(N[(N[(t$95$5 - -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[(1.0 / N[(N[Power[N[(t$95$0 * t$95$2), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] - -0.254829592), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[(t$95$6 / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(t$95$6 / t$95$4), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.254829592 - N[(N[(-0.284496736 - N[(N[(N[(N[(-1.453152027 - t$95$5), $MachinePrecision] / t$95$3), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 80.1%

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

9. Applied rewrites 80.2%

   \[\leadsto \frac{{1}^{3} - \frac{1}{\frac{{\left(e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}{{\left(\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -0.254829592\right)}^{3}}}}{\mathsf{fma}\left(\color{blue}{\frac{{\left(\frac{-0.254829592 - \frac{-0.284496736 - \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1}{\frac{-0.254829592 - \frac{-0.284496736 - \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot e^{x \cdot x}} - 1}}, \frac{-0.254829592 - \frac{-0.284496736 - \frac{\frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot e^{x \cdot x}}, 1\right)} \]
10. Add Preprocessing

Alternative 2: 80.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 80.1%

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

9. Applied rewrites 80.1%

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

11. Applied rewrites 80.1%

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

Alternative 3: 80.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(Float64(-0.254829592 - Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_1)) / t_3) - 1.421413741) / t_1)) / t_1)) / Float64(t_3 * t_0))
	return Float64(Float64((1.0 ^ 3.0) - Float64(1.0 / Float64((Float64(t_0 * t_2) ^ 3.0) / (Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_2)) / t_2) - 1.421413741) / t_2) - -0.284496736) / t_3) - -0.254829592) ^ 3.0)))) / fma(Float64(t_4 - -1.0), t_4, 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-0.254829592 - N[(N[(-0.284496736 - N[(N[(N[(N[(-1.453152027 - N[(-1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[(1.0 / N[(N[Power[N[(t$95$0 * t$95$2), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] - -0.254829592), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 - -1.0), $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 80.1%

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

Alternative 4: 79.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = fma(-0.3275911, abs(x), -1.0)
	t_2 = fma(abs(x), 0.3275911, 1.0)
	t_3 = Float64(Float64(-0.254829592 - Float64(Float64(-0.284496736 - Float64(Float64(Float64(Float64(-1.453152027 - Float64(-1.061405429 / t_2)) / t_1) - 1.421413741) / t_2)) / t_2)) / Float64(t_1 * exp(Float64(x * x))))
	return Float64(Float64((1.0 ^ 3.0) - Float64(1.0 / Float64((Float64(1.0 + Float64(0.3275911 * abs(x))) ^ 3.0) / (Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / t_0) - 1.421413741) / t_0) - -0.284496736) / t_1) - -0.254829592) ^ 3.0)))) / fma(Float64(t_3 - -1.0), t_3, 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.254829592 - N[(N[(-0.284496736 - N[(N[(N[(N[(-1.453152027 - N[(-1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[(1.0 / N[(N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] - -0.254829592), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 - -1.0), $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 80.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{{1}^{3} - \frac{1}{\frac{\color{blue}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}}{{\left(\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -0.254829592\right)}^{3}}}}{\mathsf{fma}\left(\frac{-0.254829592 - \frac{-0.284496736 - \frac{\frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot e^{x \cdot x}} - -1, \frac{-0.254829592 - \frac{-0.284496736 - \frac{\frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot e^{x \cdot x}}, 1\right)} \]
9. Step-by-step derivation

   1. lower-pow.f64 N/A

      \[\leadsto \frac{{1}^{3} - \frac{1}{\frac{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{\color{blue}{3}}}{{\left(\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-31853699}{125000000}\right)}^{3}}}}{\mathsf{fma}\left(\frac{\frac{-31853699}{125000000} - \frac{\frac{-8890523}{31250000} - \frac{\frac{\frac{-1453152027}{1000000000} - \frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot e^{x \cdot x}} - -1, \frac{\frac{-31853699}{125000000} - \frac{\frac{-8890523}{31250000} - \frac{\frac{\frac{-1453152027}{1000000000} - \frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot e^{x \cdot x}}, 1\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{{1}^{3} - \frac{1}{\frac{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}{{\left(\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-31853699}{125000000}\right)}^{3}}}}{\mathsf{fma}\left(\frac{\frac{-31853699}{125000000} - \frac{\frac{-8890523}{31250000} - \frac{\frac{\frac{-1453152027}{1000000000} - \frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot e^{x \cdot x}} - -1, \frac{\frac{-31853699}{125000000} - \frac{\frac{-8890523}{31250000} - \frac{\frac{\frac{-1453152027}{1000000000} - \frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot e^{x \cdot x}}, 1\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{1}^{3} - \frac{1}{\frac{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}{{\left(\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-31853699}{125000000}\right)}^{3}}}}{\mathsf{fma}\left(\frac{\frac{-31853699}{125000000} - \frac{\frac{-8890523}{31250000} - \frac{\frac{\frac{-1453152027}{1000000000} - \frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot e^{x \cdot x}} - -1, \frac{\frac{-31853699}{125000000} - \frac{\frac{-8890523}{31250000} - \frac{\frac{\frac{-1453152027}{1000000000} - \frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot e^{x \cdot x}}, 1\right)} \]

   4. lower-fabs.f64 79.3%

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

10. Applied rewrites 79.3%

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

Alternative 5: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.0%

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

Alternative 6: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.0%

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

Alternative 7: 79.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 79.0%

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

5. Applied rewrites 79.0%

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

Alternative 8: 79.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 79.0%

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

Alternative 9: 79.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

5. Applied rewrites 79.0%

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

Alternative 10: 79.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

Alternative 11: 77.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.4%

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

8. Applied rewrites 77.4%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-unsound-*.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-/.f64 77.4%

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

10. Applied rewrites 77.4%

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

Alternative 12: 77.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.4%

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

8. Applied rewrites 77.4%

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

10. Applied rewrites 77.4%

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

Alternative 13: 77.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.4%

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

Reproduce

herbie shell --seed 2025212 
(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)))))))