Jmat.Real.erf

Percentage Accurate: 79.1% → 86.5%

Time: 24.7s

Alternatives: 18

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: 86.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(-Float64(x * x)))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = Float64(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / Float64(t_1 * t_1)) + Float64(0.254829592 / t_1))
	t_3 = Float64(t_0 * t_2)
	t_4 = t_3 ^ 2.0
	return Float64(Float64(Float64(1.0 / Float64(1.0 + t_4)) + Float64((t_3 ^ 4.0) / Float64(-1.0 - t_4))) * Float64(1.0 / fma(t_2, t_0, 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.254829592 / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, N[(N[(N[(1.0 / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$3, 4.0], $MachinePrecision] / N[(-1.0 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$2 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.4%

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

4. Applied rewrites 79.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

6. Applied rewrites 79.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

8. Applied rewrites 86.8%

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

9. Final simplification 86.8%

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

Alternative 2: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_1}}{t\_1}}{t\_1}}{t\_1 \cdot t\_1} + \frac{0.254829592}{t\_1}\\ t_3 := t\_0 \cdot t\_2\\ \frac{1 - {t\_3}^{8}}{\left(1 + {t\_3}^{4}\right) \cdot \left(\left(1 + {t\_3}^{2}\right) \cdot \mathsf{fma}\left(t\_0, t\_2, 1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- (* x x))))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2
         (+
          (/
           (+
            -0.284496736
            (/
             (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1))
             t_1))
           (* t_1 t_1))
          (/ 0.254829592 t_1)))
        (t_3 (* t_0 t_2)))
   (/
    (- 1.0 (pow t_3 8.0))
    (* (+ 1.0 (pow t_3 4.0)) (* (+ 1.0 (pow t_3 2.0)) (fma t_0 t_2 1.0))))))

C

double code(double x) {
	double t_0 = exp(-(x * x));
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / (t_1 * t_1)) + (0.254829592 / t_1);
	double t_3 = t_0 * t_2;
	return (1.0 - pow(t_3, 8.0)) / ((1.0 + pow(t_3, 4.0)) * ((1.0 + pow(t_3, 2.0)) * fma(t_0, t_2, 1.0)));
}

Julia

function code(x)
	t_0 = exp(Float64(-Float64(x * x)))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = Float64(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / Float64(t_1 * t_1)) + Float64(0.254829592 / t_1))
	t_3 = Float64(t_0 * t_2)
	return Float64(Float64(1.0 - (t_3 ^ 8.0)) / Float64(Float64(1.0 + (t_3 ^ 4.0)) * Float64(Float64(1.0 + (t_3 ^ 2.0)) * fma(t_0, t_2, 1.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.254829592 / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$3, 8.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[Power[t$95$3, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 79.1%

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

4. Applied rewrites 79.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

6. Applied rewrites 79.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

8. Applied rewrites 86.5%

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

10. Applied rewrites 79.3%

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

11. Final simplification 79.3%

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

Reproduce

herbie shell --seed 2024228 
(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)))))))