Jmat.Real.erf

Percentage Accurate: 78.9% → 86.3%

Time: 18.5s

Alternatives: 10

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: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 86.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{x}\\ t_1 := {\left(e^{x}\right)}^{\left(-x\right)}\\ t_2 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_3 := 0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_2}}{t\_2}}{t\_2}}{t\_2}\\ t_4 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_5 := \frac{\frac{\frac{\frac{1.061405429}{t\_4} + -1.453152027}{t\_4} + 1.421413741}{t\_4} + -0.284496736}{t\_4} + 0.254829592\\ t_6 := \frac{\frac{t\_5}{t\_4}}{t\_0}\\ t_7 := \mathsf{fma}\left(\frac{t\_1}{t\_4}, t\_5, 1\right)\\ t_8 := {t\_7}^{-2}\\ t_9 := \frac{{t\_6}^{2} + 1}{t\_7}\\ \frac{{\left(\frac{t\_8}{t\_9}\right)}^{2} - {\left(\frac{{t\_6}^{4} \cdot t\_8}{t\_9}\right)}^{2}}{{t\_9}^{-1} \cdot \left(\left({\left(\frac{\frac{t\_3}{t\_0}}{t\_2}\right)}^{4} + 1\right) \cdot {\left(\mathsf{fma}\left(t\_3, \frac{t\_1}{t\_2}, 1\right)\right)}^{-2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) x))
        (t_1 (pow (exp x) (- x)))
        (t_2 (fma (fabs x) 0.3275911 1.0))
        (t_3
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_2)) t_2))
             t_2))
           t_2)))
        (t_4 (fma 0.3275911 (fabs x) 1.0))
        (t_5
         (+
          (/
           (+
            (/
             (+ (/ (+ (/ 1.061405429 t_4) -1.453152027) t_4) 1.421413741)
             t_4)
            -0.284496736)
           t_4)
          0.254829592))
        (t_6 (/ (/ t_5 t_4) t_0))
        (t_7 (fma (/ t_1 t_4) t_5 1.0))
        (t_8 (pow t_7 -2.0))
        (t_9 (/ (+ (pow t_6 2.0) 1.0) t_7)))
   (/
    (- (pow (/ t_8 t_9) 2.0) (pow (/ (* (pow t_6 4.0) t_8) t_9) 2.0))
    (*
     (pow t_9 -1.0)
     (*
      (+ (pow (/ (/ t_3 t_0) t_2) 4.0) 1.0)
      (pow (fma t_3 (/ t_1 t_2) 1.0) -2.0))))))

C

double code(double x) {
	double t_0 = pow(exp(x), x);
	double t_1 = pow(exp(x), -x);
	double t_2 = fma(fabs(x), 0.3275911, 1.0);
	double t_3 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_2)) / t_2)) / t_2)) / t_2);
	double t_4 = fma(0.3275911, fabs(x), 1.0);
	double t_5 = (((((((1.061405429 / t_4) + -1.453152027) / t_4) + 1.421413741) / t_4) + -0.284496736) / t_4) + 0.254829592;
	double t_6 = (t_5 / t_4) / t_0;
	double t_7 = fma((t_1 / t_4), t_5, 1.0);
	double t_8 = pow(t_7, -2.0);
	double t_9 = (pow(t_6, 2.0) + 1.0) / t_7;
	return (pow((t_8 / t_9), 2.0) - pow(((pow(t_6, 4.0) * t_8) / t_9), 2.0)) / (pow(t_9, -1.0) * ((pow(((t_3 / t_0) / t_2), 4.0) + 1.0) * pow(fma(t_3, (t_1 / t_2), 1.0), -2.0)));
}

Julia

function code(x)
	t_0 = exp(x) ^ x
	t_1 = exp(x) ^ Float64(-x)
	t_2 = fma(abs(x), 0.3275911, 1.0)
	t_3 = Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_2)) / t_2)) / t_2)) / t_2))
	t_4 = fma(0.3275911, abs(x), 1.0)
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_4) + -1.453152027) / t_4) + 1.421413741) / t_4) + -0.284496736) / t_4) + 0.254829592)
	t_6 = Float64(Float64(t_5 / t_4) / t_0)
	t_7 = fma(Float64(t_1 / t_4), t_5, 1.0)
	t_8 = t_7 ^ -2.0
	t_9 = Float64(Float64((t_6 ^ 2.0) + 1.0) / t_7)
	return Float64(Float64((Float64(t_8 / t_9) ^ 2.0) - (Float64(Float64((t_6 ^ 4.0) * t_8) / t_9) ^ 2.0)) / Float64((t_9 ^ -1.0) * Float64(Float64((Float64(Float64(t_3 / t_0) / t_2) ^ 4.0) + 1.0) * (fma(t_3, Float64(t_1 / t_2), 1.0) ^ -2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / 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[(N[(N[(N[(N[(N[(1.061405429 / t$95$4), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$4), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$4), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$4), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 / t$95$4), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$1 / t$95$4), $MachinePrecision] * t$95$5 + 1.0), $MachinePrecision]}, Block[{t$95$8 = N[Power[t$95$7, -2.0], $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[Power[t$95$6, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / t$95$7), $MachinePrecision]}, N[(N[(N[Power[N[(t$95$8 / t$95$9), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(N[(N[Power[t$95$6, 4.0], $MachinePrecision] * t$95$8), $MachinePrecision] / t$95$9), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$9, -1.0], $MachinePrecision] * N[(N[(N[Power[N[(N[(t$95$3 / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], 4.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[N[(t$95$3 * N[(t$95$1 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Initial program 77.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

4. Applied rewrites 77.5%

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

6. Applied rewrites 82.2%

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

8. Applied rewrites 85.6%

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

10. Applied rewrites 85.6%

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

Alternative 2: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := \frac{\frac{t\_1}{t\_0}}{{\left(e^{x}\right)}^{x}}\\ t_3 := \mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t\_0}, t\_1, 1\right)\\ t_4 := -{t\_3}^{-2}\\ t_5 := \frac{{t\_2}^{2} + 1}{t\_3}\\ \mathsf{fma}\left(t\_4, \frac{-1}{t\_5}, \frac{{t\_2}^{4} \cdot t\_4}{t\_5}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (+
          (/
           (+
            (/
             (+ (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0) 1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_2 (/ (/ t_1 t_0) (pow (exp x) x)))
        (t_3 (fma (/ (pow (exp x) (- x)) t_0) t_1 1.0))
        (t_4 (- (pow t_3 -2.0)))
        (t_5 (/ (+ (pow t_2 2.0) 1.0) t_3)))
   (fma t_4 (/ -1.0 t_5) (/ (* (pow t_2 4.0) t_4) t_5))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = (((((((1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_2 = (t_1 / t_0) / pow(exp(x), x);
	double t_3 = fma((pow(exp(x), -x) / t_0), t_1, 1.0);
	double t_4 = -pow(t_3, -2.0);
	double t_5 = (pow(t_2, 2.0) + 1.0) / t_3;
	return fma(t_4, (-1.0 / t_5), ((pow(t_2, 4.0) * t_4) / t_5));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 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)
	t_2 = Float64(Float64(t_1 / t_0) / (exp(x) ^ x))
	t_3 = fma(Float64((exp(x) ^ Float64(-x)) / t_0), t_1, 1.0)
	t_4 = Float64(-(t_3 ^ -2.0))
	t_5 = Float64(Float64((t_2 ^ 2.0) + 1.0) / t_3)
	return fma(t_4, Float64(-1.0 / t_5), Float64(Float64((t_2 ^ 4.0) * t_4) / t_5))
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[(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]}, Block[{t$95$2 = N[(N[(t$95$1 / t$95$0), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]}, Block[{t$95$4 = (-N[Power[t$95$3, -2.0], $MachinePrecision])}, Block[{t$95$5 = N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / t$95$3), $MachinePrecision]}, N[(t$95$4 * N[(-1.0 / t$95$5), $MachinePrecision] + N[(N[(N[Power[t$95$2, 4.0], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 77.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

4. Applied rewrites 77.5%

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

6. Applied rewrites 82.2%

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

8. Applied rewrites 82.5%

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

9. Final simplification 82.5%

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

Alternative 3: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := {t\_1}^{-3}\\ t_3 := {t\_1}^{2}\\ t_4 := \left(\frac{1.421413741}{t\_3} - \mathsf{fma}\left(t\_0, t\_2, \frac{0.284496736}{t\_1}\right)\right) + 0.254829592\\ t_5 := \mathsf{fma}\left({t\_4}^{2}, {t\_1}^{-2}, 1\right)\\ t_6 := {t\_1}^{-4}\\ t_7 := \left(\mathsf{fma}\left(1.421413741, t\_2, \frac{0.254829592}{t\_1}\right) - \mathsf{fma}\left(t\_0, t\_6, \frac{0.284496736}{t\_3}\right)\right) + 1\\ \mathsf{fma}\left({t\_5}^{-1}, {t\_7}^{-1}, \frac{{t\_4}^{4}}{t\_5} \cdot \frac{-t\_6}{t\_7}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.061405429 (fma -0.3275911 (fabs x) -1.0)) 1.453152027))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (pow t_1 -3.0))
        (t_3 (pow t_1 2.0))
        (t_4
         (+
          (- (/ 1.421413741 t_3) (fma t_0 t_2 (/ 0.284496736 t_1)))
          0.254829592))
        (t_5 (fma (pow t_4 2.0) (pow t_1 -2.0) 1.0))
        (t_6 (pow t_1 -4.0))
        (t_7
         (+
          (-
           (fma 1.421413741 t_2 (/ 0.254829592 t_1))
           (fma t_0 t_6 (/ 0.284496736 t_3)))
          1.0)))
   (fma
    (pow t_5 -1.0)
    (pow t_7 -1.0)
    (* (/ (pow t_4 4.0) t_5) (/ (- t_6) t_7)))))

C

double code(double x) {
	double t_0 = (1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + 1.453152027;
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = pow(t_1, -3.0);
	double t_3 = pow(t_1, 2.0);
	double t_4 = ((1.421413741 / t_3) - fma(t_0, t_2, (0.284496736 / t_1))) + 0.254829592;
	double t_5 = fma(pow(t_4, 2.0), pow(t_1, -2.0), 1.0);
	double t_6 = pow(t_1, -4.0);
	double t_7 = (fma(1.421413741, t_2, (0.254829592 / t_1)) - fma(t_0, t_6, (0.284496736 / t_3))) + 1.0;
	return fma(pow(t_5, -1.0), pow(t_7, -1.0), ((pow(t_4, 4.0) / t_5) * (-t_6 / t_7)));
}

Julia

function code(x)
	t_0 = Float64(Float64(1.061405429 / fma(-0.3275911, abs(x), -1.0)) + 1.453152027)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = t_1 ^ -3.0
	t_3 = t_1 ^ 2.0
	t_4 = Float64(Float64(Float64(1.421413741 / t_3) - fma(t_0, t_2, Float64(0.284496736 / t_1))) + 0.254829592)
	t_5 = fma((t_4 ^ 2.0), (t_1 ^ -2.0), 1.0)
	t_6 = t_1 ^ -4.0
	t_7 = Float64(Float64(fma(1.421413741, t_2, Float64(0.254829592 / t_1)) - fma(t_0, t_6, Float64(0.284496736 / t_3))) + 1.0)
	return fma((t_5 ^ -1.0), (t_7 ^ -1.0), Float64(Float64((t_4 ^ 4.0) / t_5) * Float64(Float64(-t_6) / t_7)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.453152027), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -3.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(1.421413741 / t$95$3), $MachinePrecision] - N[(t$95$0 * t$95$2 + N[(0.284496736 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[t$95$4, 2.0], $MachinePrecision] * N[Power[t$95$1, -2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$1, -4.0], $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(1.421413741 * t$95$2 + N[(0.254829592 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$6 + N[(0.284496736 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[Power[t$95$5, -1.0], $MachinePrecision] * N[Power[t$95$7, -1.0], $MachinePrecision] + N[(N[(N[Power[t$95$4, 4.0], $MachinePrecision] / t$95$5), $MachinePrecision] * N[((-t$95$6) / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Initial program 77.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

4. Applied rewrites 77.5%

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

6. Applied rewrites 82.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{\left(1 + \frac{{\left(\left(\frac{31853699}{125000000} + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}^{2}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) \cdot \left(\left(1 + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)} - \frac{{\left(\left(\frac{31853699}{125000000} + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}^{4}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\left(1 + \frac{{\left(\left(\frac{31853699}{125000000} + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}^{2}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) \cdot \left(\left(1 + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right)}} \]

9. Applied rewrites 81.5%

   \[\leadsto \color{blue}{\frac{1}{\left(1 + \left(\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) - \frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{2}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + 1\right)} - \frac{\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{4}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}}{\left(1 + \left(\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) - \frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{2}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + 1\right)}} \]

11. Applied rewrites 81.8%

    \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left({\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{2}, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1\right)\right)}^{-1}, \color{blue}{{\left(\left(\mathsf{fma}\left(1.421413741, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}, \frac{0.284496736}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right)\right) + 1\right)}^{-1}}, -\frac{{\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{4}}{\mathsf{fma}\left({\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{2}, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1\right)} \cdot \frac{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}}{\left(\mathsf{fma}\left(1.421413741, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}, \frac{0.284496736}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right)\right) + 1}\right) \]

12. Final simplification 81.8%

    \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left({\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{2}, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1\right)\right)}^{-1}, {\left(\left(\mathsf{fma}\left(1.421413741, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}, \frac{0.284496736}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right)\right) + 1\right)}^{-1}, \frac{{\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{4}}{\mathsf{fma}\left({\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{2}, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1\right)} \cdot \frac{-{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}}{\left(\mathsf{fma}\left(1.421413741, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}, \frac{0.284496736}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right)\right) + 1}\right) \]
13. Add Preprocessing

Alternative 4: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := {t\_0}^{3}\\ t_2 := {t\_0}^{2}\\ t_3 := \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027\\ t_4 := \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027\\ t_5 := {t\_0}^{4}\\ t_6 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_7 := {t\_6}^{2}\\ t_8 := {t\_6}^{-3}\\ t_9 := 0.254829592 + \left(\left(\frac{1.421413741}{t\_2} - \frac{t\_4}{t\_1}\right) - \frac{0.284496736}{t\_0}\right)\\ {\left(\mathsf{fma}\left({\left(\left(\frac{1.421413741}{t\_7} - \mathsf{fma}\left(t\_3, t\_8, \frac{0.284496736}{t\_6}\right)\right) + 0.254829592\right)}^{2}, {t\_6}^{-2}, 1\right) \cdot \left(\left(\mathsf{fma}\left(1.421413741, t\_8, \frac{0.254829592}{t\_6}\right) - \mathsf{fma}\left(t\_3, {t\_6}^{-4}, \frac{0.284496736}{t\_7}\right)\right) + 1\right)\right)}^{-1} - \frac{\frac{{t\_9}^{4}}{t\_5}}{\left(1 + \left(\left(\left(\frac{1.421413741}{t\_1} + \frac{0.254829592}{t\_0}\right) - \frac{t\_4}{t\_5}\right) - \frac{0.284496736}{t\_2}\right)\right) \cdot \left(\frac{{t\_9}^{2}}{t\_2} + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (pow t_0 3.0))
        (t_2 (pow t_0 2.0))
        (t_3 (+ (/ 1.061405429 (fma -0.3275911 (fabs x) -1.0)) 1.453152027))
        (t_4 (+ (/ 1.061405429 (fma (fabs x) -0.3275911 -1.0)) 1.453152027))
        (t_5 (pow t_0 4.0))
        (t_6 (fma (fabs x) 0.3275911 1.0))
        (t_7 (pow t_6 2.0))
        (t_8 (pow t_6 -3.0))
        (t_9
         (+
          0.254829592
          (- (- (/ 1.421413741 t_2) (/ t_4 t_1)) (/ 0.284496736 t_0)))))
   (-
    (pow
     (*
      (fma
       (pow
        (+
         (- (/ 1.421413741 t_7) (fma t_3 t_8 (/ 0.284496736 t_6)))
         0.254829592)
        2.0)
       (pow t_6 -2.0)
       1.0)
      (+
       (-
        (fma 1.421413741 t_8 (/ 0.254829592 t_6))
        (fma t_3 (pow t_6 -4.0) (/ 0.284496736 t_7)))
       1.0))
     -1.0)
    (/
     (/ (pow t_9 4.0) t_5)
     (*
      (+
       1.0
       (-
        (- (+ (/ 1.421413741 t_1) (/ 0.254829592 t_0)) (/ t_4 t_5))
        (/ 0.284496736 t_2)))
      (+ (/ (pow t_9 2.0) t_2) 1.0))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = pow(t_0, 3.0);
	double t_2 = pow(t_0, 2.0);
	double t_3 = (1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + 1.453152027;
	double t_4 = (1.061405429 / fma(fabs(x), -0.3275911, -1.0)) + 1.453152027;
	double t_5 = pow(t_0, 4.0);
	double t_6 = fma(fabs(x), 0.3275911, 1.0);
	double t_7 = pow(t_6, 2.0);
	double t_8 = pow(t_6, -3.0);
	double t_9 = 0.254829592 + (((1.421413741 / t_2) - (t_4 / t_1)) - (0.284496736 / t_0));
	return pow((fma(pow((((1.421413741 / t_7) - fma(t_3, t_8, (0.284496736 / t_6))) + 0.254829592), 2.0), pow(t_6, -2.0), 1.0) * ((fma(1.421413741, t_8, (0.254829592 / t_6)) - fma(t_3, pow(t_6, -4.0), (0.284496736 / t_7))) + 1.0)), -1.0) - ((pow(t_9, 4.0) / t_5) / ((1.0 + ((((1.421413741 / t_1) + (0.254829592 / t_0)) - (t_4 / t_5)) - (0.284496736 / t_2))) * ((pow(t_9, 2.0) / t_2) + 1.0)));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = t_0 ^ 3.0
	t_2 = t_0 ^ 2.0
	t_3 = Float64(Float64(1.061405429 / fma(-0.3275911, abs(x), -1.0)) + 1.453152027)
	t_4 = Float64(Float64(1.061405429 / fma(abs(x), -0.3275911, -1.0)) + 1.453152027)
	t_5 = t_0 ^ 4.0
	t_6 = fma(abs(x), 0.3275911, 1.0)
	t_7 = t_6 ^ 2.0
	t_8 = t_6 ^ -3.0
	t_9 = Float64(0.254829592 + Float64(Float64(Float64(1.421413741 / t_2) - Float64(t_4 / t_1)) - Float64(0.284496736 / t_0)))
	return Float64((Float64(fma((Float64(Float64(Float64(1.421413741 / t_7) - fma(t_3, t_8, Float64(0.284496736 / t_6))) + 0.254829592) ^ 2.0), (t_6 ^ -2.0), 1.0) * Float64(Float64(fma(1.421413741, t_8, Float64(0.254829592 / t_6)) - fma(t_3, (t_6 ^ -4.0), Float64(0.284496736 / t_7))) + 1.0)) ^ -1.0) - Float64(Float64((t_9 ^ 4.0) / t_5) / Float64(Float64(1.0 + Float64(Float64(Float64(Float64(1.421413741 / t_1) + Float64(0.254829592 / t_0)) - Float64(t_4 / t_5)) - Float64(0.284496736 / t_2))) * Float64(Float64((t_9 ^ 2.0) / t_2) + 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[Power[t$95$0, 3.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.453152027), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.061405429 / N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.453152027), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$0, 4.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$7 = N[Power[t$95$6, 2.0], $MachinePrecision]}, Block[{t$95$8 = N[Power[t$95$6, -3.0], $MachinePrecision]}, Block[{t$95$9 = N[(0.254829592 + N[(N[(N[(1.421413741 / t$95$2), $MachinePrecision] - N[(t$95$4 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(N[(N[Power[N[(N[(N[(1.421413741 / t$95$7), $MachinePrecision] - N[(t$95$3 * t$95$8 + N[(0.284496736 / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$6, -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(1.421413741 * t$95$8 + N[(0.254829592 / t$95$6), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[Power[t$95$6, -4.0], $MachinePrecision] + N[(0.284496736 / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[(N[Power[t$95$9, 4.0], $MachinePrecision] / t$95$5), $MachinePrecision] / N[(N[(1.0 + N[(N[(N[(N[(1.421413741 / t$95$1), $MachinePrecision] + N[(0.254829592 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 / t$95$5), $MachinePrecision]), $MachinePrecision] - N[(0.284496736 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$9, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Initial program 77.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

4. Applied rewrites 77.5%

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

6. Applied rewrites 82.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{\left(1 + \frac{{\left(\left(\frac{31853699}{125000000} + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}^{2}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) \cdot \left(\left(1 + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)} - \frac{{\left(\left(\frac{31853699}{125000000} + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}^{4}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\left(1 + \frac{{\left(\left(\frac{31853699}{125000000} + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}^{2}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) \cdot \left(\left(1 + \left(-1 \cdot \frac{\frac{1453152027}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)\right)\right) - \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right)}} \]

9. Applied rewrites 81.5%

   \[\leadsto \color{blue}{\frac{1}{\left(1 + \left(\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) - \frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{2}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + 1\right)} - \frac{\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{4}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}}{\left(1 + \left(\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) - \frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{2}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + 1\right)}} \]

11. Applied rewrites 81.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left({\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{2}, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1\right) \cdot \left(\left(\mathsf{fma}\left(1.421413741, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}, \frac{0.284496736}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right)\right) + 1\right)} - \frac{\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{4}}{\color{blue}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}}}{\left(1 + \left(\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) - \frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{2}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + 1\right)} \]

12. Final simplification 81.5%

    \[\leadsto {\left(\mathsf{fma}\left({\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}} - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right) + 0.254829592\right)}^{2}, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1\right) \cdot \left(\left(\mathsf{fma}\left(1.421413741, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-3}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) - \mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.453152027, {\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-4}, \frac{0.284496736}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right)\right) + 1\right)\right)}^{-1} - \frac{\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{4}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}}{\left(1 + \left(\left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) - \frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(0.254829592 + \left(\left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} + 1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}^{2}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + 1\right)} \]
13. Add Preprocessing

Alternative 5: 78.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

4. Applied rewrites 77.5%

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

5. Final simplification 77.5%

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

Alternative 6: 78.9% accurate, 0.2× speedup

Math

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

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))
        (t_2 (/ t_1 (* (pow (exp x) x) t_0))))
   (/
    (- 1.0 (pow t_2 3.0))
    (fma t_2 (fma (pow (exp x) (- x)) (/ t_1 t_0) 1.0) 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;
	double t_2 = t_1 / (pow(exp(x), x) * t_0);
	return (1.0 - pow(t_2, 3.0)) / fma(t_2, fma(pow(exp(x), -x), (t_1 / t_0), 1.0), 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)
	t_2 = Float64(t_1 / Float64((exp(x) ^ x) * t_0))
	return Float64(Float64(1.0 - (t_2 ^ 3.0)) / fma(t_2, fma((exp(x) ^ Float64(-x)), Float64(t_1 / t_0), 1.0), 1.0))
end

Wolfram

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

Derivation

1. Initial program 77.5%

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

4. Applied rewrites 77.5%

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

Alternative 7: 55.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

4. Applied rewrites 58.0%

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

6. Applied rewrites 58.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-fabs.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-fabs.f64 52.2

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

9. Applied rewrites 52.2%

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

    1. lift-*.f64 N/A

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

    2. lift-fabs.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. sqr-abs N/A

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

    5. lower-*.f64 52.2

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

11. Applied rewrites 52.2%

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

12. Final simplification 52.2%

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

Alternative 8: 78.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((pow((1.0 + (0.3275911 * fabs(x))), -1.0) * (0.254829592 + fma((((((1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) * -1.0), (-1.0 / (t_0 * t_0)), (-0.284496736 / 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(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0) * Float64(0.254829592 + fma(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) * -1.0), Float64(-1.0 / Float64(t_0 * t_0)), Float64(-0.284496736 / 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[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(0.254829592 + N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] * -1.0), $MachinePrecision] * N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.5%

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

4. Applied rewrites 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. sqr-abs N/A

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

   5. lift-*.f64 77.5

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

6. Applied rewrites 77.5%

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

7. Final simplification 77.5%

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

Alternative 9: 78.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 77.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

4. Applied rewrites 77.5%

   \[\leadsto 1 - \color{blue}{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

6. Applied rewrites 77.5%

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

8. Applied rewrites 77.5%

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

9. Final simplification 77.5%

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

Alternative 10: 78.9% 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 77.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

4. Applied rewrites 77.5%

   \[\leadsto 1 - \color{blue}{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 77.5

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

6. Applied rewrites 77.5%

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

7. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2024299 
(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)))))))