Jmat.Real.erf

Percentage Accurate: 79.3% → 86.6%

Time: 17.7s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 86.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \frac{\frac{1.061405429}{t\_1} + -1.453152027}{t\_1}\\ t_3 := \frac{\frac{0.254829592 + \frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, t\_2, \mathsf{fma}\left(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{t\_1}^{-2}, -0.284496736\right)}{t\_1}}{t\_1}}{{\left(e^{x}\right)}^{x}}\\ t_4 := {t\_3}^{2} + 1\\ \frac{\frac{1}{t\_4} - \frac{{t\_3}^{4}}{t\_4}}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{{t\_1}^{2}}, \mathsf{fma}\left(t\_0, t\_2, -1.421413741 \cdot t\_1\right), -0.284496736\right)}{t\_1} + 0.254829592}{t\_1}, {\left(e^{x}\right)}^{\left(-x\right)}, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (/ (+ (/ 1.061405429 t_1) -1.453152027) t_1))
        (t_3
         (/
          (/
           (+
            0.254829592
            (/
             (fma
              (fma t_0 t_2 (fma -0.4656424909693051 (fabs x) -1.421413741))
              (- (pow t_1 -2.0))
              -0.284496736)
             t_1))
           t_1)
          (pow (exp x) x)))
        (t_4 (+ (pow t_3 2.0) 1.0)))
   (/
    (- (/ 1.0 t_4) (/ (pow t_3 4.0) t_4))
    (fma
     (/
      (+
       (/
        (fma
         (/ -1.0 (pow t_1 2.0))
         (fma t_0 t_2 (* -1.421413741 t_1))
         -0.284496736)
        t_1)
       0.254829592)
      t_1)
     (pow (exp x) (- x))
     1.0))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = ((1.061405429 / t_1) + -1.453152027) / t_1;
	double t_3 = ((0.254829592 + (fma(fma(t_0, t_2, fma(-0.4656424909693051, fabs(x), -1.421413741)), -pow(t_1, -2.0), -0.284496736) / t_1)) / t_1) / pow(exp(x), x);
	double t_4 = pow(t_3, 2.0) + 1.0;
	return ((1.0 / t_4) - (pow(t_3, 4.0) / t_4)) / fma((((fma((-1.0 / pow(t_1, 2.0)), fma(t_0, t_2, (-1.421413741 * t_1)), -0.284496736) / t_1) + 0.254829592) / t_1), pow(exp(x), -x), 1.0);
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = Float64(Float64(Float64(1.061405429 / t_1) + -1.453152027) / t_1)
	t_3 = Float64(Float64(Float64(0.254829592 + Float64(fma(fma(t_0, t_2, fma(-0.4656424909693051, abs(x), -1.421413741)), Float64(-(t_1 ^ -2.0)), -0.284496736) / t_1)) / t_1) / (exp(x) ^ x))
	t_4 = Float64((t_3 ^ 2.0) + 1.0)
	return Float64(Float64(Float64(1.0 / t_4) - Float64((t_3 ^ 4.0) / t_4)) / fma(Float64(Float64(Float64(fma(Float64(-1.0 / (t_1 ^ 2.0)), fma(t_0, t_2, Float64(-1.421413741 * t_1)), -0.284496736) / t_1) + 0.254829592) / t_1), (exp(x) ^ Float64(-x)), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(0.254829592 + N[(N[(N[(t$95$0 * t$95$2 + N[(-0.4656424909693051 * N[Abs[x], $MachinePrecision] + -1.421413741), $MachinePrecision]), $MachinePrecision] * (-N[Power[t$95$1, -2.0], $MachinePrecision]) + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[t$95$3, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$4), $MachinePrecision] - N[(N[Power[t$95$3, 4.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(-1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$2 + N[(-1.421413741 * t$95$1), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$1), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. lift-/.f64 N/A

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

   6. un-div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. frac-2neg N/A

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.2%

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

8. Applied rewrites 86.6%

   \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \mathsf{fma}\left(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \mathsf{fma}\left(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}\right)}^{4}}{{\left(\frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \mathsf{fma}\left(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}\right)}^{2} + 1}}}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -1.421413741 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right), -0.284496736\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, {\left(e^{x}\right)}^{\left(-x\right)}, 1\right)} \]

9. Final simplification 86.6%

   \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{0.254829592 + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \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(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, -0.284496736\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}}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{0.254829592 + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \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(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, -0.284496736\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}}\right)}^{4}}{{\left(\frac{\frac{0.254829592 + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \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(-0.4656424909693051, \left|x\right|, -1.421413741\right)\right), -{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, -0.284496736\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}}\right)}^{2} + 1}}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\frac{-1}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right), \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 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right), -0.284496736\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, {\left(e^{x}\right)}^{\left(-x\right)}, 1\right)} \]
10. Add Preprocessing

Alternative 2: 79.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{\frac{1.421413741 + \frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0}}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := \frac{\frac{t\_1}{{\left(e^{x}\right)}^{x}}}{t\_0}\\ \frac{1 - {t\_2}^{9}}{\left({t\_2}^{3} + \left({t\_2}^{6} + 1\right)\right) \cdot \mathsf{fma}\left(t\_2, \mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t\_0}, t\_1, 1\right), 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (+
          (/
           (+
            (/
             (+ 1.421413741 (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0))
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_2 (/ (/ t_1 (pow (exp x) x)) t_0)))
   (/
    (- 1.0 (pow t_2 9.0))
    (*
     (+ (pow t_2 3.0) (+ (pow t_2 6.0) 1.0))
     (fma t_2 (fma (/ (pow (exp x) (- x)) t_0) t_1 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = ((((1.421413741 + (((1.061405429 / t_0) + -1.453152027) / t_0)) / 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, 9.0)) / ((pow(t_2, 3.0) + (pow(t_2, 6.0) + 1.0)) * fma(t_2, fma((pow(exp(x), -x) / t_0), t_1, 1.0), 1.0));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(1.421413741 + Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0)) / t_0) + -0.284496736) / t_0) + 0.254829592)
	t_2 = Float64(Float64(t_1 / (exp(x) ^ x)) / t_0)
	return Float64(Float64(1.0 - (t_2 ^ 9.0)) / Float64(Float64((t_2 ^ 3.0) + Float64((t_2 ^ 6.0) + 1.0)) * fma(t_2, fma(Float64((exp(x) ^ Float64(-x)) / t_0), t_1, 1.0), 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(1.421413741 + N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$2, 9.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[(N[Power[t$95$2, 6.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.2%

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.3%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\frac{0.254829592 + \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)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{3}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \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)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{6} + {\left(\frac{0.254829592 + \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)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{3}\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)} \]

8. Applied rewrites 79.3%

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592, 1\right), 1\right) \cdot \left(\left({\left(\frac{\frac{\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{6} + 1\right) + {\left(\frac{\frac{\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}} \]

9. Final simplification 79.3%

   \[\leadsto \frac{1 - {\left(\frac{\frac{\frac{\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)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{\left({\left(\frac{\frac{\frac{\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)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3} + \left({\left(\frac{\frac{\frac{\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)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{6} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{\frac{\frac{\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)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{\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)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592, 1\right), 1\right)} \]
10. Add Preprocessing

Alternative 3: 79.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := {\left(e^{x}\right)}^{x}\\ t_3 := \frac{t\_1}{t\_0 \cdot t\_2}\\ \frac{\frac{1 - {t\_3}^{6}}{{t\_3}^{3} + 1}}{\left(\frac{\frac{t\_1}{t\_0}}{t\_2} + 1\right) + {t\_3}^{2}} \end{array} \end{array} \]

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 (pow (exp x) x))
        (t_3 (/ t_1 (* t_0 t_2))))
   (/
    (/ (- 1.0 (pow t_3 6.0)) (+ (pow t_3 3.0) 1.0))
    (+ (+ (/ (/ t_1 t_0) t_2) 1.0) (pow t_3 2.0)))))

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 = pow(exp(x), x);
	double t_3 = t_1 / (t_0 * t_2);
	return ((1.0 - pow(t_3, 6.0)) / (pow(t_3, 3.0) + 1.0)) / ((((t_1 / t_0) / t_2) + 1.0) + pow(t_3, 2.0));
}

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 = exp(x) ^ x
	t_3 = Float64(t_1 / Float64(t_0 * t_2))
	return Float64(Float64(Float64(1.0 - (t_3 ^ 6.0)) / Float64((t_3 ^ 3.0) + 1.0)) / Float64(Float64(Float64(Float64(t_1 / t_0) / t_2) + 1.0) + (t_3 ^ 2.0)))
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[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$3, 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$3, 3.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 79.2%

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.3%

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

7. Final simplification 79.3%

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

Alternative 4: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0}\\ t_2 := t\_1 + 0.254829592\\ t_3 := {\left(e^{x}\right)}^{x}\\ t_4 := t\_0 \cdot t\_3\\ \frac{\mathsf{fma}\left(-1, {\left(\frac{\frac{1}{t\_1 - 0.254829592} \cdot \left({t\_1}^{2} - 0.06493812095888646\right)}{t\_4}\right)}^{3}, 1\right)}{\left(\frac{\frac{t\_2}{t\_0}}{t\_3} + 1\right) + {\left(\frac{t\_2}{t\_4}\right)}^{2}} \end{array} \end{array} \]

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))
        (t_2 (+ t_1 0.254829592))
        (t_3 (pow (exp x) x))
        (t_4 (* t_0 t_3)))
   (/
    (fma
     -1.0
     (pow
      (/
       (* (/ 1.0 (- t_1 0.254829592)) (- (pow t_1 2.0) 0.06493812095888646))
       t_4)
      3.0)
     1.0)
    (+ (+ (/ (/ t_2 t_0) t_3) 1.0) (pow (/ t_2 t_4) 2.0)))))

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;
	double t_2 = t_1 + 0.254829592;
	double t_3 = pow(exp(x), x);
	double t_4 = t_0 * t_3;
	return fma(-1.0, pow((((1.0 / (t_1 - 0.254829592)) * (pow(t_1, 2.0) - 0.06493812095888646)) / t_4), 3.0), 1.0) / ((((t_2 / t_0) / t_3) + 1.0) + pow((t_2 / t_4), 2.0));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0)
	t_2 = Float64(t_1 + 0.254829592)
	t_3 = exp(x) ^ x
	t_4 = Float64(t_0 * t_3)
	return Float64(fma(-1.0, (Float64(Float64(Float64(1.0 / Float64(t_1 - 0.254829592)) * Float64((t_1 ^ 2.0) - 0.06493812095888646)) / t_4) ^ 3.0), 1.0) / Float64(Float64(Float64(Float64(t_2 / t_0) / t_3) + 1.0) + (Float64(t_2 / t_4) ^ 2.0)))
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[(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]}, Block[{t$95$2 = N[(t$95$1 + 0.254829592), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$3), $MachinePrecision]}, N[(N[(-1.0 * N[Power[N[(N[(N[(1.0 / N[(t$95$1 - 0.254829592), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] - 0.06493812095888646), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(t$95$2 / t$95$0), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] + N[Power[N[(t$95$2 / t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 79.2%

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.2%

   \[\leadsto \frac{\mathsf{fma}\left(-1, {\left(\frac{\color{blue}{\left({\left(\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{2} - 0.06493812095888646\right) \cdot \frac{1}{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 0.254829592}}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}, 1\right)}{{\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{2} + \left(1 - \frac{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x}} \cdot 1\right)} \]

7. Final simplification 79.2%

   \[\leadsto \frac{\mathsf{fma}\left(-1, {\left(\frac{\frac{1}{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 0.254829592} \cdot \left({\left(\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{2} - 0.06493812095888646\right)}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{3}, 1\right)}{\left(\frac{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x}} + 1\right) + {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{2}} \]
8. Add Preprocessing

Alternative 5: 79.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 79.2

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

4. Applied rewrites 79.2%

   \[\leadsto 1 - \color{blue}{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 79.2

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

6. Applied rewrites 79.2%

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

8. Applied rewrites 79.2%

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

9. Final simplification 79.2%

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

Alternative 6: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 79.2%

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

5. Final simplification 79.2%

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

Alternative 7: 79.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

4. Applied rewrites 77.5%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\left(0.0809383927946537 - {\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{2}\right) \cdot 1\right) \cdot \frac{1}{\left(-0.284496736 - \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

6. Applied rewrites 79.2%

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

7. Final simplification 79.2%

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

Alternative 8: 79.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(Float64(0.3275911 * abs(x)) - -1.0)) * 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)) * exp(Float64(Float64(-abs(x)) * abs(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[(1.0 / N[(N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision] * N[Exp[N[((-N[Abs[x], $MachinePrecision]) * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 79.2%

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

5. Final simplification 79.2%

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

Alternative 9: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 79.2

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-neg.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-exp.f64 79.2

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

   9. lift-neg.f64 N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. lift-neg.f64 N/A

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

   13. lower-*.f64 79.2

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

8. Applied rewrites 79.2%

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

9. Final simplification 79.2%

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

Alternative 10: 54.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 79.2

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.2%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f64 52.7

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

9. Applied rewrites 52.7%

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

10. Final simplification 52.7%

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

Alternative 11: 77.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.2%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
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 79.2

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 79.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 77.4%

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

10. Final simplification 77.4%

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

Reproduce

herbie shell --seed 2024308 
(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)))))))