exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 12.1s

Alternatives: 24

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x + -1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (+ (* x x) -1.0)))

C

double code(double x) {
	return exp(((x * x) + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(((x * x) + (-1.0d0)))
end function

Java

public static double code(double x) {
	return Math.exp(((x * x) + -1.0));
}

Python

def code(x):
	return math.exp(((x * x) + -1.0))

Julia

function code(x)
	return exp(Float64(Float64(x * x) + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = exp(((x * x) + -1.0));
end

Wolfram

code[x_] := N[Exp[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \frac{x \cdot \left(0.125 + \left(t\_0 \cdot t\_0\right) \cdot 0.004629629629629629\right)}{0.25 + x \cdot \left(x \cdot -0.08333333333333333\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (* x x) 0.001)
     (*
      (/ 1.0 E)
      (+
       1.0
       (*
        x
        (*
         x
         (+
          1.0
          (*
           x
           (/
            (* x (+ 0.125 (* (* t_0 t_0) 0.004629629629629629)))
            (+ 0.25 (* x (* x -0.08333333333333333))))))))))
     (exp (* x x)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / ((double) M_E)) * (1.0 + (x * (x * (1.0 + (x * ((x * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (x * (x * -0.08333333333333333)))))))));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / Math.E) * (1.0 + (x * (x * (1.0 + (x * ((x * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (x * (x * -0.08333333333333333)))))))));
	} else {
		tmp = Math.exp((x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 / math.e) * (1.0 + (x * (x * (1.0 + (x * ((x * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (x * (x * -0.08333333333333333)))))))))
	else:
		tmp = math.exp((x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.125 + Float64(Float64(t_0 * t_0) * 0.004629629629629629))) / Float64(0.25 + Float64(x * Float64(x * -0.08333333333333333))))))))));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 / 2.71828182845904523536) * (1.0 + (x * (x * (1.0 + (x * ((x * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (x * (x * -0.08333333333333333)))))))));
	else
		tmp = exp((x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(N[(x * N[(0.125 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(x * N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \color{blue}{\frac{\left(0.125 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right) \cdot x}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5\right)}}\right)\right)\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{-1}{12} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(\frac{-1}{12} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{12} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{216}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{12}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 99.9%

       \[\leadsto \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \frac{\left(0.125 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right) \cdot x}{0.25 + \color{blue}{x \cdot \left(x \cdot -0.08333333333333333\right)}}\right)\right)\right) \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \frac{x \cdot \left(0.125 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right)}{0.25 + x \cdot \left(x \cdot -0.08333333333333333\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ t_1 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \frac{\left(0.015625 - 2.143347050754458 \cdot 10^{-5} \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \frac{x}{0.25 + t\_1 \cdot \left(t\_1 + -0.5\right)}}{0.125 + t\_0 \cdot -0.004629629629629629}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x (* x (* x x)))))
        (t_1 (* x (* x 0.16666666666666666))))
   (if (<= (* x x) 4e+100)
     (*
      (/ 1.0 E)
      (+
       1.0
       (*
        x
        (*
         x
         (+
          1.0
          (*
           x
           (/
            (*
             (- 0.015625 (* 2.143347050754458e-5 (* t_0 t_0)))
             (/ x (+ 0.25 (* t_1 (+ t_1 -0.5)))))
            (+ 0.125 (* t_0 -0.004629629629629629)))))))))
     (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E)))))

C

double code(double x) {
	double t_0 = (x * x) * (x * (x * (x * x)));
	double t_1 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 4e+100) {
		tmp = (1.0 / ((double) M_E)) * (1.0 + (x * (x * (1.0 + (x * (((0.015625 - (2.143347050754458e-5 * (t_0 * t_0))) * (x / (0.25 + (t_1 * (t_1 + -0.5))))) / (0.125 + (t_0 * -0.004629629629629629))))))));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * (x * (x * x)));
	double t_1 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 4e+100) {
		tmp = (1.0 / Math.E) * (1.0 + (x * (x * (1.0 + (x * (((0.015625 - (2.143347050754458e-5 * (t_0 * t_0))) * (x / (0.25 + (t_1 * (t_1 + -0.5))))) / (0.125 + (t_0 * -0.004629629629629629))))))));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (x * (x * (x * x)))
	t_1 = x * (x * 0.16666666666666666)
	tmp = 0
	if (x * x) <= 4e+100:
		tmp = (1.0 / math.e) * (1.0 + (x * (x * (1.0 + (x * (((0.015625 - (2.143347050754458e-5 * (t_0 * t_0))) * (x / (0.25 + (t_1 * (t_1 + -0.5))))) / (0.125 + (t_0 * -0.004629629629629629))))))))
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))
	t_1 = Float64(x * Float64(x * 0.16666666666666666))
	tmp = 0.0
	if (Float64(x * x) <= 4e+100)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(Float64(Float64(0.015625 - Float64(2.143347050754458e-5 * Float64(t_0 * t_0))) * Float64(x / Float64(0.25 + Float64(t_1 * Float64(t_1 + -0.5))))) / Float64(0.125 + Float64(t_0 * -0.004629629629629629)))))))));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (x * (x * (x * x)));
	t_1 = x * (x * 0.16666666666666666);
	tmp = 0.0;
	if ((x * x) <= 4e+100)
		tmp = (1.0 / 2.71828182845904523536) * (1.0 + (x * (x * (1.0 + (x * (((0.015625 - (2.143347050754458e-5 * (t_0 * t_0))) * (x / (0.25 + (t_1 * (t_1 + -0.5))))) / (0.125 + (t_0 * -0.004629629629629629))))))));
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e+100], N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(N[(N[(0.015625 - N[(2.143347050754458e-5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(0.25 + N[(t$95$1 * N[(t$95$1 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.125 + N[(t$95$0 * -0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000006e100

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 87.5%

      \[\leadsto \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \color{blue}{\frac{\left(0.125 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right) \cdot x}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5\right)}}\right)\right)\right) \]

   11. Applied egg-rr 93.4%

       \[\leadsto \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \color{blue}{\frac{\left(0.015625 - 2.143347050754458 \cdot 10^{-5} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \cdot \frac{x}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -0.5\right)}}{0.125 + -0.004629629629629629 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}\right)\right)\right) \]

   if 4.00000000000000006e100 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \frac{\left(0.015625 - 2.143347050754458 \cdot 10^{-5} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \cdot \frac{x}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -0.5\right)}}{0.125 + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot -0.004629629629629629}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\\ t_1 := -1 - t\_0\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{1}{e} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(1 + t\_0\right)\right) \cdot t\_1\right)\right)}{1 + \left(x \cdot x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (+ (* x (* x 0.16666666666666666)) 0.5))))
        (t_1 (- -1.0 t_0)))
   (if (<= (* x x) 4e+100)
     (/
      (* (/ 1.0 E) (+ 1.0 (* (* x x) (* (* (* x x) (+ 1.0 t_0)) t_1))))
      (+ 1.0 (* (* x x) t_1)))
     (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E)))))

C

double code(double x) {
	double t_0 = x * (x * ((x * (x * 0.16666666666666666)) + 0.5));
	double t_1 = -1.0 - t_0;
	double tmp;
	if ((x * x) <= 4e+100) {
		tmp = ((1.0 / ((double) M_E)) * (1.0 + ((x * x) * (((x * x) * (1.0 + t_0)) * t_1)))) / (1.0 + ((x * x) * t_1));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * ((x * (x * 0.16666666666666666)) + 0.5));
	double t_1 = -1.0 - t_0;
	double tmp;
	if ((x * x) <= 4e+100) {
		tmp = ((1.0 / Math.E) * (1.0 + ((x * x) * (((x * x) * (1.0 + t_0)) * t_1)))) / (1.0 + ((x * x) * t_1));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * ((x * (x * 0.16666666666666666)) + 0.5))
	t_1 = -1.0 - t_0
	tmp = 0
	if (x * x) <= 4e+100:
		tmp = ((1.0 / math.e) * (1.0 + ((x * x) * (((x * x) * (1.0 + t_0)) * t_1)))) / (1.0 + ((x * x) * t_1))
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(Float64(x * Float64(x * 0.16666666666666666)) + 0.5)))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (Float64(x * x) <= 4e+100)
		tmp = Float64(Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(1.0 + t_0)) * t_1)))) / Float64(1.0 + Float64(Float64(x * x) * t_1)));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * ((x * (x * 0.16666666666666666)) + 0.5));
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if ((x * x) <= 4e+100)
		tmp = ((1.0 / 2.71828182845904523536) * (1.0 + ((x * x) * (((x * x) * (1.0 + t_0)) * t_1)))) / (1.0 + ((x * x) * t_1));
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e+100], N[(N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000006e100

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right)}{\color{blue}{1 - x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{\mathsf{E}\left(\right)} \cdot \left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)}{\color{blue}{1 - x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{E}\left(\right)} \cdot \left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 - x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right)}\right) \]

   9. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{e} \cdot \left(1 - \left(x \cdot x\right) \cdot \left(\left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}} \]

   if 4.00000000000000006e100 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{1}{e} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right) \cdot \left(-1 - x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(-1 - x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{-1 + \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_1\right)}{-1 + t\_1}}{e}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* x x) (+ (* x (* x 0.16666666666666666)) 0.5))))
        (t_1 (* (* x x) t_0)))
   (if (<= (* x x) 4e+100)
     (/ (/ (+ -1.0 (* (* x x) (* t_0 t_1))) (+ -1.0 t_1)) E)
     (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E)))))

C

double code(double x) {
	double t_0 = 1.0 + ((x * x) * ((x * (x * 0.16666666666666666)) + 0.5));
	double t_1 = (x * x) * t_0;
	double tmp;
	if ((x * x) <= 4e+100) {
		tmp = ((-1.0 + ((x * x) * (t_0 * t_1))) / (-1.0 + t_1)) / ((double) M_E);
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 1.0 + ((x * x) * ((x * (x * 0.16666666666666666)) + 0.5));
	double t_1 = (x * x) * t_0;
	double tmp;
	if ((x * x) <= 4e+100) {
		tmp = ((-1.0 + ((x * x) * (t_0 * t_1))) / (-1.0 + t_1)) / Math.E;
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + ((x * x) * ((x * (x * 0.16666666666666666)) + 0.5))
	t_1 = (x * x) * t_0
	tmp = 0
	if (x * x) <= 4e+100:
		tmp = ((-1.0 + ((x * x) * (t_0 * t_1))) / (-1.0 + t_1)) / math.e
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * Float64(x * 0.16666666666666666)) + 0.5)))
	t_1 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (Float64(x * x) <= 4e+100)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(x * x) * Float64(t_0 * t_1))) / Float64(-1.0 + t_1)) / exp(1));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + ((x * x) * ((x * (x * 0.16666666666666666)) + 0.5));
	t_1 = (x * x) * t_0;
	tmp = 0.0;
	if ((x * x) <= 4e+100)
		tmp = ((-1.0 + ((x * x) * (t_0 * t_1))) / (-1.0 + t_1)) / 2.71828182845904523536;
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e+100], N[(N[(N[(-1.0 + N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000006e100

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

   9. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{e}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) + 1\right), \mathsf{E.f64}\left(\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right) - 1 \cdot 1}{\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) - 1}\right), \mathsf{E.f64}\left(\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right) - 1 \cdot 1\right), \left(\left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) - 1\right)\right), \mathsf{E.f64}\left(\right)\right) \]

   11. Applied egg-rr 93.3%

       \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right) - 1}{\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) - 1}}}{e} \]

   if 4.00000000000000006e100 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{-1 + \left(x \cdot x\right) \cdot \left(\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}{-1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \frac{\left(x \cdot x\right) \cdot \left(1 - t\_0 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\right)\right)}{1 - \left(x \cdot x\right) \cdot t\_0}}{e}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* x (* x 0.16666666666666666)) 0.5)))
   (if (<= (* x x) 5e+153)
     (/
      (+
       1.0
       (/
        (* (* x x) (- 1.0 (* t_0 (* (* x (* x (* x x))) t_0))))
        (- 1.0 (* (* x x) t_0))))
      E)
     (* (* x x) (* 0.5 (/ (* x x) E))))))

C

double code(double x) {
	double t_0 = (x * (x * 0.16666666666666666)) + 0.5;
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 + (((x * x) * (1.0 - (t_0 * ((x * (x * (x * x))) * t_0)))) / (1.0 - ((x * x) * t_0)))) / ((double) M_E);
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (x * (x * 0.16666666666666666)) + 0.5;
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 + (((x * x) * (1.0 - (t_0 * ((x * (x * (x * x))) * t_0)))) / (1.0 - ((x * x) * t_0)))) / Math.E;
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * (x * 0.16666666666666666)) + 0.5
	tmp = 0
	if (x * x) <= 5e+153:
		tmp = (1.0 + (((x * x) * (1.0 - (t_0 * ((x * (x * (x * x))) * t_0)))) / (1.0 - ((x * x) * t_0)))) / math.e
	else:
		tmp = (x * x) * (0.5 * ((x * x) / math.e))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * Float64(x * 0.16666666666666666)) + 0.5)
	tmp = 0.0
	if (Float64(x * x) <= 5e+153)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(1.0 - Float64(t_0 * Float64(Float64(x * Float64(x * Float64(x * x))) * t_0)))) / Float64(1.0 - Float64(Float64(x * x) * t_0)))) / exp(1));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(Float64(x * x) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * (x * 0.16666666666666666)) + 0.5;
	tmp = 0.0;
	if ((x * x) <= 5e+153)
		tmp = (1.0 + (((x * x) * (1.0 - (t_0 * ((x * (x * (x * x))) * t_0)))) / (1.0 - ((x * x) * t_0)))) / 2.71828182845904523536;
	else
		tmp = (x * x) * (0.5 * ((x * x) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+153], N[(N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e153

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{e}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(x \cdot x\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{1 - x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)}\right)\right), \mathsf{E.f64}\left(\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \left(1 - x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

   11. Applied egg-rr 91.4%

       \[\leadsto \frac{1 + \color{blue}{\frac{\left(1 - \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot x\right)}{1 - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)}}}{e} \]

   if 5.00000000000000018e153 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{1}{2} \cdot \frac{{x}^{4}}{e^{1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\color{blue}{e^{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}}{e^{1}} \]

      4. pow-sqr N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)}{e^{1}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}}{e^{\color{blue}{1}}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2}}{e^{1}} \cdot \color{blue}{{x}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2}}{e^{1}} \cdot {x}^{2} \]

      8. associate-*r/ N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2}}{e^{1}}\right) \cdot {\color{blue}{x}}^{2} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{e^{1}}\right) \cdot {x}^{2} \]

      10. associate-*r/ N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{e^{1}}\right)\right) \cdot {x}^{2} \]

      11. exp-1-e N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right) \cdot {x}^{2} \]

      12. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{\mathsf{E}\left(\right)} \cdot {x}^{2}\right)}\right)\right) \]

      18. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{e^{1}} \cdot {x}^{2}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{e^{1}}}\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{e^{\color{blue}{1}}}\right)\right) \]

      21. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \frac{\left(x \cdot x\right) \cdot \left(1 - \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.125 + \left(t\_0 \cdot t\_0\right) \cdot 0.004629629629629629\right)}{0.25 + t\_1 \cdot \left(t\_1 - 0.5\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* x (* x 0.16666666666666666))))
   (if (<= (* x x) 5e+153)
     (*
      (/ 1.0 E)
      (+
       1.0
       (*
        x
        (*
         x
         (+
          1.0
          (/
           (* (* x x) (+ 0.125 (* (* t_0 t_0) 0.004629629629629629)))
           (+ 0.25 (* t_1 (- t_1 0.5)))))))))
     (* (* x x) (* 0.5 (/ (* x x) E))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 / ((double) M_E)) * (1.0 + (x * (x * (1.0 + (((x * x) * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (t_1 * (t_1 - 0.5))))))));
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 / Math.E) * (1.0 + (x * (x * (1.0 + (((x * x) * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (t_1 * (t_1 - 0.5))))))));
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = x * (x * 0.16666666666666666)
	tmp = 0
	if (x * x) <= 5e+153:
		tmp = (1.0 / math.e) * (1.0 + (x * (x * (1.0 + (((x * x) * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (t_1 * (t_1 - 0.5))))))))
	else:
		tmp = (x * x) * (0.5 * ((x * x) / math.e))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * Float64(x * 0.16666666666666666))
	tmp = 0.0
	if (Float64(x * x) <= 5e+153)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.125 + Float64(Float64(t_0 * t_0) * 0.004629629629629629))) / Float64(0.25 + Float64(t_1 * Float64(t_1 - 0.5)))))))));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(Float64(x * x) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = x * (x * 0.16666666666666666);
	tmp = 0.0;
	if ((x * x) <= 5e+153)
		tmp = (1.0 / 2.71828182845904523536) * (1.0 + (x * (x * (1.0 + (((x * x) * (0.125 + ((t_0 * t_0) * 0.004629629629629629))) / (0.25 + (t_1 * (t_1 - 0.5))))))));
	else
		tmp = (x * x) * (0.5 * ((x * x) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+153], N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$1 * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e153

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{{\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 89.0%

      \[\leadsto \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(0.125 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5\right)}}\right)\right)\right) \]

   if 5.00000000000000018e153 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{1}{2} \cdot \frac{{x}^{4}}{e^{1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\color{blue}{e^{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}}{e^{1}} \]

      4. pow-sqr N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)}{e^{1}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}}{e^{\color{blue}{1}}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2}}{e^{1}} \cdot \color{blue}{{x}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2}}{e^{1}} \cdot {x}^{2} \]

      8. associate-*r/ N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2}}{e^{1}}\right) \cdot {\color{blue}{x}}^{2} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{e^{1}}\right) \cdot {x}^{2} \]

      10. associate-*r/ N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{e^{1}}\right)\right) \cdot {x}^{2} \]

      11. exp-1-e N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right) \cdot {x}^{2} \]

      12. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{\mathsf{E}\left(\right)} \cdot {x}^{2}\right)}\right)\right) \]

      18. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{e^{1}} \cdot {x}^{2}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{e^{1}}}\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{e^{\color{blue}{1}}}\right)\right) \]

      21. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(1 + \frac{x \cdot \left(x \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.004629629629629629\right)\right)\right)\right)\right)}{0.25 + t\_0 \cdot \left(t\_0 + -0.5\right)}\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.16666666666666666))))
   (if (<= (* x x) 5e+153)
     (/
      (+
       1.0
       (*
        (* x x)
        (+
         1.0
         (/
          (*
           x
           (*
            x
            (+
             0.125
             (* (* x x) (* x (* x (* (* x x) 0.004629629629629629)))))))
          (+ 0.25 (* t_0 (+ t_0 -0.5)))))))
      E)
     (* (* x x) (* 0.5 (/ (* x x) E))))))

C

double code(double x) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 + ((x * x) * (1.0 + ((x * (x * (0.125 + ((x * x) * (x * (x * ((x * x) * 0.004629629629629629))))))) / (0.25 + (t_0 * (t_0 + -0.5))))))) / ((double) M_E);
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 + ((x * x) * (1.0 + ((x * (x * (0.125 + ((x * x) * (x * (x * ((x * x) * 0.004629629629629629))))))) / (0.25 + (t_0 * (t_0 + -0.5))))))) / Math.E;
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * 0.16666666666666666)
	tmp = 0
	if (x * x) <= 5e+153:
		tmp = (1.0 + ((x * x) * (1.0 + ((x * (x * (0.125 + ((x * x) * (x * (x * ((x * x) * 0.004629629629629629))))))) / (0.25 + (t_0 * (t_0 + -0.5))))))) / math.e
	else:
		tmp = (x * x) * (0.5 * ((x * x) / math.e))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.16666666666666666))
	tmp = 0.0
	if (Float64(x * x) <= 5e+153)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * Float64(x * Float64(0.125 + Float64(Float64(x * x) * Float64(x * Float64(x * Float64(Float64(x * x) * 0.004629629629629629))))))) / Float64(0.25 + Float64(t_0 * Float64(t_0 + -0.5))))))) / exp(1));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(Float64(x * x) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * 0.16666666666666666);
	tmp = 0.0;
	if ((x * x) <= 5e+153)
		tmp = (1.0 + ((x * x) * (1.0 + ((x * (x * (0.125 + ((x * x) * (x * (x * ((x * x) * 0.004629629629629629))))))) / (0.25 + (t_0 * (t_0 + -0.5))))))) / 2.71828182845904523536;
	else
		tmp = (x * x) * (0.5 * ((x * x) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+153], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * N[(x * N[(0.125 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$0 * N[(t$95$0 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e153

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{e}} \]

   11. Applied egg-rr 89.0%

       \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(1 + \color{blue}{\frac{x \cdot \left(x \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.004629629629629629\right)\right)\right)\right)\right)}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -0.5\right)}}\right)}{e} \]

   if 5.00000000000000018e153 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{1}{2} \cdot \frac{{x}^{4}}{e^{1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\color{blue}{e^{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}}{e^{1}} \]

      4. pow-sqr N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)}{e^{1}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}}{e^{\color{blue}{1}}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2}}{e^{1}} \cdot \color{blue}{{x}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2}}{e^{1}} \cdot {x}^{2} \]

      8. associate-*r/ N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2}}{e^{1}}\right) \cdot {\color{blue}{x}}^{2} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{e^{1}}\right) \cdot {x}^{2} \]

      10. associate-*r/ N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{e^{1}}\right)\right) \cdot {x}^{2} \]

      11. exp-1-e N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right) \cdot {x}^{2} \]

      12. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{\mathsf{E}\left(\right)} \cdot {x}^{2}\right)}\right)\right) \]

      18. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{e^{1}} \cdot {x}^{2}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{e^{1}}}\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{e^{\color{blue}{1}}}\right)\right) \]

      21. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \frac{x \cdot x}{2}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(1 + \frac{t\_0 \cdot t\_0}{8}\right)}{1 + t\_1 \cdot \left(-1 + t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (/ (* x x) 2.0)))
   (if (<= (* x x) 5e+153)
     (*
      (/ 1.0 E)
      (+
       1.0
       (/
        (* (* x x) (+ 1.0 (/ (* t_0 t_0) 8.0)))
        (+ 1.0 (* t_1 (+ -1.0 t_1))))))
     (* (* x x) (* 0.5 (/ (* x x) E))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) / 2.0;
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 / ((double) M_E)) * (1.0 + (((x * x) * (1.0 + ((t_0 * t_0) / 8.0))) / (1.0 + (t_1 * (-1.0 + t_1)))));
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) / 2.0;
	double tmp;
	if ((x * x) <= 5e+153) {
		tmp = (1.0 / Math.E) * (1.0 + (((x * x) * (1.0 + ((t_0 * t_0) / 8.0))) / (1.0 + (t_1 * (-1.0 + t_1)))));
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (x * x) / 2.0
	tmp = 0
	if (x * x) <= 5e+153:
		tmp = (1.0 / math.e) * (1.0 + (((x * x) * (1.0 + ((t_0 * t_0) / 8.0))) / (1.0 + (t_1 * (-1.0 + t_1)))))
	else:
		tmp = (x * x) * (0.5 * ((x * x) / math.e))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) / 2.0)
	tmp = 0.0
	if (Float64(x * x) <= 5e+153)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(t_0 * t_0) / 8.0))) / Float64(1.0 + Float64(t_1 * Float64(-1.0 + t_1))))));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(Float64(x * x) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (x * x) / 2.0;
	tmp = 0.0;
	if ((x * x) <= 5e+153)
		tmp = (1.0 / 2.71828182845904523536) * (1.0 + (((x * x) * (1.0 + ((t_0 * t_0) / 8.0))) / (1.0 + (t_1 * (-1.0 + t_1)))));
	else
		tmp = (x * x) * (0.5 * ((x * x) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+153], N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000018e153

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 78.7%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{{1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)}}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)}\right)\right)\right) \]

   9. Applied egg-rr 88.7%

      \[\leadsto \frac{1}{e} \cdot \left(1 + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(1 + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{8}\right)}{1 + \frac{x \cdot x}{2} \cdot \left(\frac{x \cdot x}{2} - 1\right)}}\right) \]

   if 5.00000000000000018e153 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{1}{2} \cdot \frac{{x}^{4}}{e^{1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\color{blue}{e^{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}}{e^{1}} \]

      4. pow-sqr N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)}{e^{1}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}}{e^{\color{blue}{1}}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2}}{e^{1}} \cdot \color{blue}{{x}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2}}{e^{1}} \cdot {x}^{2} \]

      8. associate-*r/ N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2}}{e^{1}}\right) \cdot {\color{blue}{x}}^{2} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{e^{1}}\right) \cdot {x}^{2} \]

      10. associate-*r/ N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{e^{1}}\right)\right) \cdot {x}^{2} \]

      11. exp-1-e N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right) \cdot {x}^{2} \]

      12. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{\mathsf{E}\left(\right)} \cdot {x}^{2}\right)}\right)\right) \]

      18. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{e^{1}} \cdot {x}^{2}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{e^{1}}}\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{e^{\color{blue}{1}}}\right)\right) \]

      21. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)}\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(1 + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{8}\right)}{1 + \frac{x \cdot x}{2} \cdot \left(-1 + \frac{x \cdot x}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} + \frac{x \cdot x}{e} \cdot \left(1 + \frac{x \cdot x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001)
   (+ (/ 1.0 E) (* (/ (* x x) E) (+ 1.0 (/ (* x x) 2.0))))
   (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / ((double) M_E)) + (((x * x) / ((double) M_E)) * (1.0 + ((x * x) / 2.0)));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / Math.E) + (((x * x) / Math.E) * (1.0 + ((x * x) / 2.0)));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 / math.e) + (((x * x) / math.e) * (1.0 + ((x * x) / 2.0)))
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 / exp(1)) + Float64(Float64(Float64(x * x) / exp(1)) * Float64(1.0 + Float64(Float64(x * x) / 2.0))));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 / 2.71828182845904523536) + (((x * x) / 2.71828182845904523536) * (1.0 + ((x * x) / 2.0)));
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 / E), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot 1 + \color{blue}{\frac{1}{\mathsf{E}\left(\right)} \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{1}{\mathsf{E}\left(\right)} + \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{\mathsf{E}\left(\right)}\right), \color{blue}{\left(\frac{1}{\mathsf{E}\left(\right)} \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left(\color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      5. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left(\frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left(\frac{1}{\mathsf{E}\left(\right)} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left(\left(\frac{1}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right)\right), \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{\mathsf{E}\left(\right)}\right), \left(\color{blue}{1} + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(\frac{x \cdot x}{\mathsf{E}\left(\right)}\right), \left(\color{blue}{1} + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), \mathsf{E}\left(\right)\right), \left(\color{blue}{1} + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E}\left(\right)\right), \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \]

      13. E-lowering-E.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      17. div-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \frac{x}{\color{blue}{2}}\right)\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \left(\frac{x \cdot x}{\color{blue}{2}}\right)\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{2}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), 2\right)\right)\right)\right) \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{e} + \frac{x \cdot x}{e} \cdot \left(1 + \frac{x \cdot x}{2}\right)} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 83.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 92.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001)
   (* (/ 1.0 E) (+ 1.0 (* x (* x (+ 1.0 (* (* x x) 0.5))))))
   (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / ((double) M_E)) * (1.0 + (x * (x * (1.0 + ((x * x) * 0.5)))));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / Math.E) * (1.0 + (x * (x * (1.0 + ((x * x) * 0.5)))));
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 / math.e) * (1.0 + (x * (x * (1.0 + ((x * x) * 0.5)))))
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(Float64(x * x) * 0.5))))));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 / 2.71828182845904523536) * (1.0 + (x * (x * (1.0 + ((x * x) * 0.5)))));
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 83.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 92.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 E)
  (+
   1.0
   (* x (* x (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.16666666666666666))))))))))

C

double code(double x) {
	return (1.0 / ((double) M_E)) * (1.0 + (x * (x * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))));
}

Java

public static double code(double x) {
	return (1.0 / Math.E) * (1.0 + (x * (x * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))));
}

Python

def code(x):
	return (1.0 / math.e) * (1.0 + (x * (x * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))))

Julia

function code(x)
	return Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666)))))))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.71828182845904523536) * (1.0 + (x * (x * (1.0 + (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))));
end

Wolfram

code[x_] := N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

7. Simplified 91.2%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 13: 92.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + \frac{x \cdot x}{2}\right)\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001)
   (/ (+ 1.0 (* x (* x (+ 1.0 (/ (* x x) 2.0))))) E)
   (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) / 2.0))))) / ((double) M_E);
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) / 2.0))))) / Math.E;
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) / 2.0))))) / math.e
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(Float64(x * x) / 2.0))))) / exp(1));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) / 2.0))))) / 2.71828182845904523536;
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + \left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{x}{2}\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{x \cdot x}{2}\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot x\right), 2\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), 2\right)\right)\right)\right)\right), \mathsf{E}\left(\right)\right) \]

      14. E-lowering-E.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), 2\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(x \cdot \left(1 + \frac{x \cdot x}{2}\right)\right)}{e}} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 83.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 91.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(x \cdot x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001)
   (* (/ 1.0 E) (+ (* x x) 1.0))
   (* 0.16666666666666666 (/ (* (* x x) (* (* x x) (* x x))) E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / ((double) M_E)) * ((x * x) + 1.0);
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / Math.E) * ((x * x) + 1.0);
	} else {
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 / math.e) * ((x * x) + 1.0)
	else:
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(Float64(x * x) + 1.0));
	else
		tmp = Float64(0.16666666666666666 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 / 2.71828182845904523536) * ((x * x) + 1.0);
	else
		tmp = 0.16666666666666666 * (((x * x) * ((x * x) * (x * x))) / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 / E), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

      2. *-commutative N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      7. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      8. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

      11. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

   7. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{6}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{\mathsf{E}\left(\right)} \]

      2. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \frac{{x}^{6} \cdot 1}{e^{1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \color{blue}{\frac{1}{e^{1}}}\right) \]

      4. exp-1-e N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{6} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)}\right) \]

      6. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{6} \cdot \frac{1}{e^{1}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6} \cdot 1}{\color{blue}{e^{1}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\frac{{x}^{6}}{e^{\color{blue}{1}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left({x}^{6}\right), \color{blue}{\left(e^{1}\right)}\right)\right) \]

   10. Simplified 83.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{e}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 92.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}{e} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   1.0
   (* x (* x (+ 1.0 (* x (* x (+ (* x (* x 0.16666666666666666)) 0.5)))))))
  E))

C

double code(double x) {
	return (1.0 + (x * (x * (1.0 + (x * (x * ((x * (x * 0.16666666666666666)) + 0.5))))))) / ((double) M_E);
}

Java

public static double code(double x) {
	return (1.0 + (x * (x * (1.0 + (x * (x * ((x * (x * 0.16666666666666666)) + 0.5))))))) / Math.E;
}

Python

def code(x):
	return (1.0 + (x * (x * (1.0 + (x * (x * ((x * (x * 0.16666666666666666)) + 0.5))))))) / math.e

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(Float64(x * Float64(x * 0.16666666666666666)) + 0.5))))))) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (x * (x * (1.0 + (x * (x * ((x * (x * 0.16666666666666666)) + 0.5))))))) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(x * N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

7. Simplified 91.2%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

9. Applied egg-rr 91.2%

   \[\leadsto \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{e}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}, \mathsf{E.f64}\left(\right)\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot x\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    15. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{6} \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    18. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    19. *-lowering-*.f64 91.2%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

12. Simplified 91.2%

    \[\leadsto \frac{\color{blue}{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)}}{e} \]

13. Final simplification 91.2%

    \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}{e} \]
14. Add Preprocessing

Alternative 16: 92.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 E)
  (+ 1.0 (* x (* x (+ 1.0 (* x (* (* x x) (* x 0.16666666666666666)))))))))

C

double code(double x) {
	return (1.0 / ((double) M_E)) * (1.0 + (x * (x * (1.0 + (x * ((x * x) * (x * 0.16666666666666666)))))));
}

Java

public static double code(double x) {
	return (1.0 / Math.E) * (1.0 + (x * (x * (1.0 + (x * ((x * x) * (x * 0.16666666666666666)))))));
}

Python

def code(x):
	return (1.0 / math.e) * (1.0 + (x * (x * (1.0 + (x * ((x * x) * (x * 0.16666666666666666)))))))

Julia

function code(x)
	return Float64(Float64(1.0 / exp(1)) * Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * x) * Float64(x * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.71828182845904523536) * (1.0 + (x * (x * (1.0 + (x * ((x * x) * (x * 0.16666666666666666)))))));
end

Wolfram

code[x_] := N[(N[(1.0 / E), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

7. Simplified 91.2%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {x}^{3}\right)}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 90.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 90.9%

    \[\leadsto \frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\right)}\right)\right)\right) \]
11. Add Preprocessing

Alternative 17: 87.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(x \cdot x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001)
   (* (/ 1.0 E) (+ (* x x) 1.0))
   (* (/ (* x x) E) (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / ((double) M_E)) * ((x * x) + 1.0);
	} else {
		tmp = ((x * x) / ((double) M_E)) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / Math.E) * ((x * x) + 1.0);
	} else {
		tmp = ((x * x) / Math.E) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 / math.e) * ((x * x) + 1.0)
	else:
		tmp = ((x * x) / math.e) * (1.0 + ((x * x) * 0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(Float64(x * x) + 1.0));
	else
		tmp = Float64(Float64(Float64(x * x) / exp(1)) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 / 2.71828182845904523536) * ((x * x) + 1.0);
	else
		tmp = ((x * x) / 2.71828182845904523536) * (1.0 + ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 / E), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

      2. *-commutative N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      7. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      8. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

      11. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)} + \frac{1}{{x}^{2} \cdot \mathsf{E}\left(\right)}\right)} \]

   10. Simplified 75.5%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{x \cdot x}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(x \cdot x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{e} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 92.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right)}{e} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ 1.0 (* (* x x) (+ 1.0 (* (* x (* x (* x x))) 0.16666666666666666)))) E))

C

double code(double x) {
	return (1.0 + ((x * x) * (1.0 + ((x * (x * (x * x))) * 0.16666666666666666)))) / ((double) M_E);
}

Java

public static double code(double x) {
	return (1.0 + ((x * x) * (1.0 + ((x * (x * (x * x))) * 0.16666666666666666)))) / Math.E;
}

Python

def code(x):
	return (1.0 + ((x * x) * (1.0 + ((x * (x * (x * x))) * 0.16666666666666666)))) / math.e

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * Float64(x * Float64(x * x))) * 0.16666666666666666)))) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((x * x) * (1.0 + ((x * (x * (x * x))) * 0.16666666666666666)))) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot e^{-1}\right) + \frac{1}{2} \cdot e^{-1}\right)\right)} \]

7. Simplified 91.2%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

9. Applied egg-rr 91.2%

   \[\leadsto \color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{e}} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {x}^{4}\right)}\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{4}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    3. pow-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    7. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot {x}^{3}\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left({x}^{3}\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    9. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

    13. *-lowering-*.f64 90.9%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right), \mathsf{E.f64}\left(\right)\right) \]

12. Simplified 90.9%

    \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(1 + \color{blue}{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)}{e} \]

13. Final simplification 90.9%

    \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right)}{e} \]
14. Add Preprocessing

Alternative 19: 87.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e} \cdot \left(x \cdot x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001)
   (* (/ 1.0 E) (+ (* x x) 1.0))
   (* (* x x) (* 0.5 (/ (* x x) E)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / ((double) M_E)) * ((x * x) + 1.0);
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (1.0 / Math.E) * ((x * x) + 1.0);
	} else {
		tmp = (x * x) * (0.5 * ((x * x) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (1.0 / math.e) * ((x * x) + 1.0)
	else:
		tmp = (x * x) * (0.5 * ((x * x) / math.e))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(1.0 / exp(1)) * Float64(Float64(x * x) + 1.0));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(Float64(x * x) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (1.0 / 2.71828182845904523536) * ((x * x) + 1.0);
	else
		tmp = (x * x) * (0.5 * ((x * x) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(1.0 / E), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

      2. *-commutative N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      7. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      8. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

      11. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(e^{-1} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{e^{-1}}\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot \left(\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \color{blue}{e^{-1}}\right) \]

      3. associate-*r* N/A

         \[\leadsto e^{-1} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{e^{-1}} \]

      4. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \cdot e^{-1} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto e^{-1} \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} \cdot 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto e^{-1} \cdot \left(1 + \left({x}^{2} + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      9. associate-+l+ N/A

         \[\leadsto e^{-1} \cdot \left(\left(1 + {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto e^{-1} \cdot \left(\left({x}^{2} + 1\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left(\left({x}^{2} + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{4}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{1}{2} \cdot \frac{{x}^{4}}{e^{1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{4}}{\color{blue}{e^{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{\left(2 \cdot 2\right)}}{e^{1}} \]

      4. pow-sqr N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)}{e^{1}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot {x}^{2}}{e^{\color{blue}{1}}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2}}{e^{1}} \cdot \color{blue}{{x}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2}}{e^{1}} \cdot {x}^{2} \]

      8. associate-*r/ N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2}}{e^{1}}\right) \cdot {\color{blue}{x}}^{2} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{e^{1}}\right) \cdot {x}^{2} \]

      10. associate-*r/ N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{e^{1}}\right)\right) \cdot {x}^{2} \]

      11. exp-1-e N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right) \cdot {x}^{2} \]

      12. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{\mathsf{E}\left(\right)} \cdot {x}^{2}\right)}\right)\right) \]

      18. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{e^{1}} \cdot {x}^{2}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{e^{1}}}\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{e^{\color{blue}{1}}}\right)\right) \]

      21. exp-1-e N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)}\right)\right) \]

   10. Simplified 75.5%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot \frac{x \cdot x}{e}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 76.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001) (/ 1.0 E) (/ 1.0 (/ E (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = 1.0 / (((double) M_E) / (x * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = 1.0 / (Math.E / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = 1.0 / math.e
	else:
		tmp = 1.0 / (math.e / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(1.0 / Float64(exp(1) / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = 1.0 / (2.71828182845904523536 / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(1.0 / E), $MachinePrecision], N[(1.0 / N[(E / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(1\right)} \]

      2. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{e^{1}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{1}\right)}\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right) \]

      5. E-lowering-E.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

      2. *-commutative N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      7. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      8. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

      11. *-lowering-*.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{{x}^{2}}{e^{1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2}\right), \color{blue}{\left(e^{1}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \left(e^{\color{blue}{1}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(e^{\color{blue}{1}}\right)\right) \]

      5. exp-1-e N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E}\left(\right)\right) \]

      6. E-lowering-E.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right) \]

   10. Simplified 57.0%

       \[\leadsto \color{blue}{\frac{x \cdot x}{e}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{E}\left(\right)}{x \cdot x}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\mathsf{E}\left(\right)}{x \cdot x}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{E}\left(\right), \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       4. E-lowering-E.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{E.f64}\left(\right), \left(\color{blue}{x} \cdot x\right)\right)\right) \]

       5. *-lowering-*.f64 57.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{E.f64}\left(\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   12. Applied egg-rr 57.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{e}{x \cdot x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 76.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.001) (/ 1.0 E) (* x (/ x E))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = x * (x / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = x * (x / Math.E);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = 1.0 / math.e
	else:
		tmp = x * (x / math.e)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(x * Float64(x / exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = x * (x / 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(1.0 / E), $MachinePrecision], N[(x * N[(x / E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(1\right)} \]

      2. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{e^{1}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{1}\right)}\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right) \]

      5. E-lowering-E.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   6. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

      2. *-commutative N/A

         \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      7. exp-1-e N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      8. E-lowering-E.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

      11. *-lowering-*.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

   7. Simplified 57.0%

      \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{\mathsf{E}\left(\right)}} \]
   9. Step-by-step derivation

      1. exp-1-e N/A

         \[\leadsto \frac{{x}^{2}}{e^{1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{2}\right), \color{blue}{\left(e^{1}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \left(e^{\color{blue}{1}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(e^{\color{blue}{1}}\right)\right) \]

      5. exp-1-e N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E}\left(\right)\right) \]

      6. E-lowering-E.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{E.f64}\left(\right)\right) \]

   10. Simplified 57.0%

       \[\leadsto \color{blue}{\frac{x \cdot x}{e}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto x \cdot \color{blue}{\frac{x}{\mathsf{E}\left(\right)}} \]

       2. *-lft-identity N/A

          \[\leadsto x \cdot \frac{1 \cdot x}{\mathsf{E}\left(\right)} \]

       3. associate-*l/ N/A

          \[\leadsto x \cdot \left(\frac{1}{\mathsf{E}\left(\right)} \cdot \color{blue}{x}\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\frac{1}{\mathsf{E}\left(\right)} \cdot x\right) \cdot \color{blue}{x} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\mathsf{E}\left(\right)} \cdot x\right), \color{blue}{x}\right) \]

       6. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot x}{\mathsf{E}\left(\right)}\right), x\right) \]

       7. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{\mathsf{E}\left(\right)}\right), x\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{E}\left(\right)\right), x\right) \]

       9. E-lowering-E.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{E.f64}\left(\right)\right), x\right) \]

   12. Applied egg-rr 57.0%

       \[\leadsto \color{blue}{\frac{x}{e} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 76.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \cdot \left(x \cdot x + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 E) (+ (* x x) 1.0)))

C

double code(double x) {
	return (1.0 / ((double) M_E)) * ((x * x) + 1.0);
}

Java

public static double code(double x) {
	return (1.0 / Math.E) * ((x * x) + 1.0);
}

Python

def code(x):
	return (1.0 / math.e) * ((x * x) + 1.0)

Julia

function code(x)
	return Float64(Float64(1.0 / exp(1)) * Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.71828182845904523536) * ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(N[(1.0 / E), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
6. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

   2. *-commutative N/A

      \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

   5. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

   7. exp-1-e N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

   8. E-lowering-E.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

   11. *-lowering-*.f64 77.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

7. Simplified 77.1%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]
8. Add Preprocessing

Alternative 23: 76.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x + 1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ (* x x) 1.0) E))

C

double code(double x) {
	return ((x * x) + 1.0) / ((double) M_E);
}

Java

public static double code(double x) {
	return ((x * x) + 1.0) / Math.E;
}

Python

def code(x):
	return ((x * x) + 1.0) / math.e

Julia

function code(x)
	return Float64(Float64(Float64(x * x) + 1.0) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) + 1.0) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
6. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{e^{-1}} \]

   2. *-commutative N/A

      \[\leadsto e^{-1} \cdot \color{blue}{\left({x}^{2} + 1\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{-1}\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(1\right)}\right), \left({\color{blue}{x}}^{2} + 1\right)\right) \]

   5. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{e^{1}}\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{1}\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

   7. exp-1-e N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

   8. E-lowering-E.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left({x}^{2}\right), \color{blue}{1}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right) \]

   11. *-lowering-*.f64 77.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right) \]

7. Simplified 77.1%

   \[\leadsto \color{blue}{\frac{1}{e} \cdot \left(x \cdot x + 1\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot x + 1\right) \cdot \color{blue}{\frac{1}{\mathsf{E}\left(\right)}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{x \cdot x + 1}{\color{blue}{\mathsf{E}\left(\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x + 1\right), \color{blue}{\mathsf{E}\left(\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + x \cdot x\right), \mathsf{E}\left(\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot x\right)\right), \mathsf{E}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{E}\left(\right)\right) \]

   7. E-lowering-E.f64 77.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{E.f64}\left(\right)\right) \]

9. Applied egg-rr 77.1%

   \[\leadsto \color{blue}{\frac{1 + x \cdot x}{e}} \]

10. Final simplification 77.1%

    \[\leadsto \frac{x \cdot x + 1}{e} \]
11. Add Preprocessing

Alternative 24: 50.9% accurate, 35.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 E))

C

double code(double x) {
	return 1.0 / ((double) M_E);
}

Java

public static double code(double x) {
	return 1.0 / Math.E;
}

Python

def code(x):
	return 1.0 / math.e

Julia

function code(x)
	return Float64(1.0 / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(1.0 / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto e^{\mathsf{neg}\left(1\right)} \]

   2. rec-exp N/A

      \[\leadsto \frac{1}{\color{blue}{e^{1}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{1}\right)}\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{E}\left(\right)\right) \]

   5. E-lowering-E.f64 48.6%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{E.f64}\left(\right)\right) \]

7. Simplified 48.6%

   \[\leadsto \color{blue}{\frac{1}{e}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))