Jmat.Real.erf

Percentage Accurate: 79.1% → 98.9%

Time: 50.1s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    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: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 98.7%

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

8. Applied rewrites 98.7%

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 2: 86.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := {\left(e^{x}\right)}^{x}\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_2} + 1.421413741}{t\_2}}{t\_2} + 0.254829592}{t\_1 \cdot t\_2}\\ t_4 := {t\_3}^{2} + 1\\ {\left(\frac{1 + \frac{\frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_0}}{t\_1}}{{t\_4}^{-1} - \frac{{t\_3}^{4}}{t\_4}}\right)}^{-1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (pow (exp x) x))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3
         (/
          (+
           (/
            (+
             -0.284496736
             (/
              (+
               (/
                (+
                 (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0))
                 -1.453152027)
                t_2)
               1.421413741)
              t_2))
            t_2)
           0.254829592)
          (* t_1 t_2)))
        (t_4 (+ (pow t_3 2.0) 1.0)))
   (pow
    (/
     (+
      1.0
      (/
       (/
        (+
         (/
          (+
           (/ (+ (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0) 1.421413741) t_0)
           -0.284496736)
          t_0)
         0.254829592)
        t_0)
       t_1))
     (- (pow t_4 -1.0) (/ (pow t_3 4.0) t_4)))
    -1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = pow(exp(x), x);
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = (((-0.284496736 + (((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_2) + 1.421413741) / t_2)) / t_2) + 0.254829592) / (t_1 * t_2);
	double t_4 = pow(t_3, 2.0) + 1.0;
	return pow(((1.0 + ((((((((((1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) / t_1)) / (pow(t_4, -1.0) - (pow(t_3, 4.0) / t_4))), -1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = exp(x) ^ x
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = Float64(Float64(Float64(Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_2) + 1.421413741) / t_2)) / t_2) + 0.254829592) / Float64(t_1 * t_2))
	t_4 = Float64((t_3 ^ 2.0) + 1.0)
	return Float64(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) / t_1)) / Float64((t_4 ^ -1.0) - Float64((t_3 ^ 4.0) / 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[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(-0.284496736 + N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[t$95$3, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[Power[N[(N[(1.0 + N[(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] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$4, -1.0], $MachinePrecision] - N[(N[Power[t$95$3, 4.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 86.2%

   \[\leadsto \frac{1}{\frac{1 - \frac{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x}}}{\color{blue}{\frac{1}{{\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}}} \]

7. Final simplification 86.2%

   \[\leadsto {\left(\frac{1 + \frac{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x}}}{{\left({\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1\right)}^{-1} - \frac{{\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}\right)}^{-1} \]
8. Add Preprocessing

Alternative 3: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_0} + 1.421413741}{t\_0}}{t\_0} + 0.254829592\\ t_2 := \frac{t\_1}{{\left(e^{x}\right)}^{x} \cdot t\_0}\\ \frac{1 - {t\_2}^{6}}{\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t\_0}, t\_1, 1\right) \cdot \left(\left({t\_2}^{4} + {t\_2}^{2}\right) + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (+
          (/
           (+
            -0.284496736
            (/
             (+
              (/
               (+ (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0)) -1.453152027)
               t_0)
              1.421413741)
             t_0))
           t_0)
          0.254829592))
        (t_2 (/ t_1 (* (pow (exp x) x) t_0))))
   (/
    (- 1.0 (pow t_2 6.0))
    (*
     (fma (/ (pow (exp x) (- x)) t_0) t_1 1.0)
     (+ (+ (pow t_2 4.0) (pow t_2 2.0)) 1.0)))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = ((-0.284496736 + (((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0)) / t_0) + 0.254829592;
	double t_2 = t_1 / (pow(exp(x), x) * t_0);
	return (1.0 - pow(t_2, 6.0)) / (fma((pow(exp(x), -x) / t_0), t_1, 1.0) * ((pow(t_2, 4.0) + pow(t_2, 2.0)) + 1.0));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0)) / t_0) + 0.254829592)
	t_2 = Float64(t_1 / Float64((exp(x) ^ x) * t_0))
	return Float64(Float64(1.0 - (t_2 ^ 6.0)) / Float64(fma(Float64((exp(x) ^ Float64(-x)) / t_0), t_1, 1.0) * Float64(Float64((t_2 ^ 4.0) + (t_2 ^ 2.0)) + 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.284496736 + N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$2, 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[(N[(N[Power[t$95$2, 4.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{6}}{\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592, 1\right) \cdot \left(\left({\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4} + {\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}\right) + 1\right)}} \]
7. Add Preprocessing

Alternative 4: 79.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.8%

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

8. Applied rewrites 78.8%

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

9. Final simplification 78.8%

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

Alternative 5: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_0} + 1.421413741}{t\_0}}{t\_0} + 0.254829592\\ t_2 := \frac{t\_1}{{\left(e^{x}\right)}^{x} \cdot t\_0}\\ \frac{1 - {t\_2}^{4}}{\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t\_0}, t\_1, 1\right) \cdot \left({t\_2}^{2} + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (+
          (/
           (+
            -0.284496736
            (/
             (+
              (/
               (+ (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0)) -1.453152027)
               t_0)
              1.421413741)
             t_0))
           t_0)
          0.254829592))
        (t_2 (/ t_1 (* (pow (exp x) x) t_0))))
   (/
    (- 1.0 (pow t_2 4.0))
    (* (fma (/ (pow (exp x) (- x)) t_0) t_1 1.0) (+ (pow t_2 2.0) 1.0)))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = ((-0.284496736 + (((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0)) / t_0) + 0.254829592;
	double t_2 = t_1 / (pow(exp(x), x) * t_0);
	return (1.0 - pow(t_2, 4.0)) / (fma((pow(exp(x), -x) / t_0), t_1, 1.0) * (pow(t_2, 2.0) + 1.0));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0)) / t_0) + 0.254829592)
	t_2 = Float64(t_1 / Float64((exp(x) ^ x) * t_0))
	return Float64(Float64(1.0 - (t_2 ^ 4.0)) / Float64(fma(Float64((exp(x) ^ Float64(-x)) / t_0), t_1, 1.0) * Float64((t_2 ^ 2.0) + 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.284496736 + N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[(N[Power[t$95$2, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592, 1\right) \cdot \left({\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1\right)}} \]
7. Add Preprocessing

Alternative 6: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{0.254829592 + \frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{t\_0}}{t\_0} + -0.284496736}{t\_0}}{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)} \cdot \frac{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}{{\left(e^{x}\right)}^{x}}\\ \frac{1 - {t\_1}^{2}}{1 + t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (*
          (/
           (+
            0.254829592
            (/
             (+
              (/
               (+
                1.421413741
                (/
                 (+
                  -1.453152027
                  (/ -1.061405429 (fma (fabs x) -0.3275911 -1.0)))
                 t_0))
               t_0)
              -0.284496736)
             t_0))
           (fma (* x x) 0.10731592879921 -1.0))
          (/ (fma 0.3275911 (fabs x) -1.0) (pow (exp x) x)))))
   (/ (- 1.0 (pow t_1 2.0)) (+ 1.0 t_1))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = ((0.254829592 + ((((1.421413741 + ((-1.453152027 + (-1.061405429 / fma(fabs(x), -0.3275911, -1.0))) / t_0)) / t_0) + -0.284496736) / t_0)) / fma((x * x), 0.10731592879921, -1.0)) * (fma(0.3275911, fabs(x), -1.0) / pow(exp(x), x));
	return (1.0 - pow(t_1, 2.0)) / (1.0 + t_1);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(Float64(0.254829592 + Float64(Float64(Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(-1.061405429 / fma(abs(x), -0.3275911, -1.0))) / t_0)) / t_0) + -0.284496736) / t_0)) / fma(Float64(x * x), 0.10731592879921, -1.0)) * Float64(fma(0.3275911, abs(x), -1.0) / (exp(x) ^ x)))
	return Float64(Float64(1.0 - (t_1 ^ 2.0)) / Float64(1.0 + t_1))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.254829592 + N[(N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(-1.061405429 / N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * 0.10731592879921 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

8. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)} \cdot \frac{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}{{\left(e^{x}\right)}^{x}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)} \cdot \frac{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}{{\left(e^{x}\right)}^{x}}}} \]
9. Add Preprocessing

Alternative 7: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ \frac{1 - {\left(\frac{t\_1}{{\left(e^{x}\right)}^{x} \cdot t\_0}\right)}^{2}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{t\_1}{t\_0}, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1
         (+
          (/
           (+
            (/
             (+ (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0) 1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592)))
   (/
    (- 1.0 (pow (/ t_1 (* (pow (exp x) x) t_0)) 2.0))
    (fma (pow (exp x) (- x)) (/ t_1 t_0) 1.0))))

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592)
	return Float64(Float64(1.0 - (Float64(t_1 / Float64((exp(x) ^ x) * t_0)) ^ 2.0)) / fma((exp(x) ^ Float64(-x)), Float64(t_1 / t_0), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(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] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(1.0 - N[Power[N[(t$95$1 / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

Alternative 8: 79.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

   1. lift-+.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\color{blue}{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. lift-/.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\color{blue}{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. frac-2neg N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)\right)}} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. lift-fma.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right| + 1\right)}\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. +-commutative N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\left(1 + \frac{3275911}{10000000} \cdot \color{blue}{\left|x\right|}\right)\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\left(1 + \frac{3275911}{10000000} \cdot \color{blue}{\left|x\right|}\right)\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. +-commutative N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right| + 1\right)}\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   9. *-commutative N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1\right)\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   10. lift-fma.f64 N/A

       \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   11. lift-neg.f64 N/A

       \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)}{\color{blue}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   12. div-inv N/A

       \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   13. lower-fma.f64 N/A

       \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}\right)\right), \frac{1}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{1421413741}{1000000000}\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

8. Applied rewrites 78.7%

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

9. Final simplification 78.7%

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

Alternative 9: 79.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

   1. lift-*.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   2. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\color{blue}{\left|x\right|} \cdot \left|x\right|} \]

   3. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\left|x\right| \cdot \color{blue}{\left|x\right|}} \]

   4. sqr-abs N/A

      \[\leadsto 1 - \left(\frac{\frac{\frac{-8890523}{31250000} + \frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, -1\right)\right) \cdot e^{-\color{blue}{x \cdot x}} \]

   5. lift-*.f64 78.7

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

8. Applied rewrites 78.7%

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

9. Final simplification 78.7%

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

Alternative 10: 79.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. lift-neg.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lift-neg.f64 N/A

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

   8. frac-2neg N/A

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

   9. lift-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

6. Applied rewrites 78.7%

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

7. Final simplification 78.7%

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

Alternative 11: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs N/A

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

   5. lift-*.f64 78.7

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

6. Applied rewrites 78.7%

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

7. Final simplification 78.7%

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

Alternative 12: 55.6% accurate, 262.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 78.7%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
8. Step-by-step derivation

9. Applied rewrites 54.9%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(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)))))))