exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 11.4s

Alternatives: 17

Speedup: 1.0×

• 49.7% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.5\\ t_1 := \left(x \cdot x\right) \cdot \left(-1 - t\_0\right)\\ t_2 := 1 + t\_0\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1 + \left(x \cdot t\_2\right) \cdot \left(\left(t\_2 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(x \cdot x\right) \cdot t\_1\right)\right)\right)}{\left(1 + t\_2 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot t\_2\right)\right)\right) \cdot \left(1 + t\_1\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.5)) (t_1 (* (* x x) (- -1.0 t_0))) (t_2 (+ 1.0 t_0)))
   (if (<= (* x x) 5e-8)
     (/
      (/
       (+ 1.0 (* (* x t_2) (* (* t_2 (* x (* x x))) (* t_2 (* (* x x) t_1)))))
       (* (+ 1.0 (* t_2 (* (* x x) (* (* x x) t_2)))) (+ 1.0 t_1)))
      E)
     (exp (* x x)))))

C

double code(double x) {
	double t_0 = (x * x) * 0.5;
	double t_1 = (x * x) * (-1.0 - t_0);
	double t_2 = 1.0 + t_0;
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = ((1.0 + ((x * t_2) * ((t_2 * (x * (x * x))) * (t_2 * ((x * x) * t_1))))) / ((1.0 + (t_2 * ((x * x) * ((x * x) * t_2)))) * (1.0 + t_1))) / ((double) M_E);
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (x * x) * 0.5;
	double t_1 = (x * x) * (-1.0 - t_0);
	double t_2 = 1.0 + t_0;
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = ((1.0 + ((x * t_2) * ((t_2 * (x * (x * x))) * (t_2 * ((x * x) * t_1))))) / ((1.0 + (t_2 * ((x * x) * ((x * x) * t_2)))) * (1.0 + t_1))) / Math.E;
	} else {
		tmp = Math.exp((x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * 0.5
	t_1 = (x * x) * (-1.0 - t_0)
	t_2 = 1.0 + t_0
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = ((1.0 + ((x * t_2) * ((t_2 * (x * (x * x))) * (t_2 * ((x * x) * t_1))))) / ((1.0 + (t_2 * ((x * x) * ((x * x) * t_2)))) * (1.0 + t_1))) / math.e
	else:
		tmp = math.exp((x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.5)
	t_1 = Float64(Float64(x * x) * Float64(-1.0 - t_0))
	t_2 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x * t_2) * Float64(Float64(t_2 * Float64(x * Float64(x * x))) * Float64(t_2 * Float64(Float64(x * x) * t_1))))) / Float64(Float64(1.0 + Float64(t_2 * Float64(Float64(x * x) * Float64(Float64(x * x) * t_2)))) * Float64(1.0 + t_1))) / exp(1));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * 0.5;
	t_1 = (x * x) * (-1.0 - t_0);
	t_2 = 1.0 + t_0;
	tmp = 0.0;
	if ((x * x) <= 5e-8)
		tmp = ((1.0 + ((x * t_2) * ((t_2 * (x * (x * x))) * (t_2 * ((x * x) * t_1))))) / ((1.0 + (t_2 * ((x * x) * ((x * x) * t_2)))) * (1.0 + t_1))) / 2.71828182845904523536;
	else
		tmp = exp((x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], N[(N[(N[(1.0 + N[(N[(x * t$95$2), $MachinePrecision] * N[(N[(t$95$2 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 4.9999999999999998e-8 < (*.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 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.7%

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

Alternative 3: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ t_2 := \left(x \cdot x\right) \cdot \left(-1 - t\_1\right)\\ t_3 := 1 + t\_1\\ t_4 := x \cdot t\_1\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{1 + \left(x \cdot t\_3\right) \cdot \left(\left(t\_3 \cdot t\_0\right) \cdot \left(t\_3 \cdot \left(\left(x \cdot x\right) \cdot t\_2\right)\right)\right)}{\left(1 + t\_3 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot t\_3\right)\right)\right) \cdot \left(1 + t\_2\right)}}{e}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{x \cdot \left(t\_0 + t\_0 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.125\right)\right)}{x \cdot x + t\_4 \cdot \left(t\_4 - x\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* (* x x) 0.5))
        (t_2 (* (* x x) (- -1.0 t_1)))
        (t_3 (+ 1.0 t_1))
        (t_4 (* x t_1)))
   (if (<= (* x x) 2e+45)
     (/
      (/
       (+ 1.0 (* (* x t_3) (* (* t_3 t_0) (* t_3 (* (* x x) t_2)))))
       (* (+ 1.0 (* t_3 (* (* x x) (* (* x x) t_3)))) (+ 1.0 t_2)))
      E)
     (if (<= (* x x) 5e+102)
       (/
        (+
         1.0
         (/
          (* x (+ t_0 (* t_0 (* (* (* x x) (* (* x x) (* x x))) 0.125))))
          (+ (* x x) (* t_4 (- t_4 x)))))
        E)
       (+
        1.0
        (*
         (* x x)
         (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.16666666666666666))))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * 0.5;
	double t_2 = (x * x) * (-1.0 - t_1);
	double t_3 = 1.0 + t_1;
	double t_4 = x * t_1;
	double tmp;
	if ((x * x) <= 2e+45) {
		tmp = ((1.0 + ((x * t_3) * ((t_3 * t_0) * (t_3 * ((x * x) * t_2))))) / ((1.0 + (t_3 * ((x * x) * ((x * x) * t_3)))) * (1.0 + t_2))) / ((double) M_E);
	} else if ((x * x) <= 5e+102) {
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_4 * (t_4 - x))))) / ((double) M_E);
	} else {
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * 0.5;
	double t_2 = (x * x) * (-1.0 - t_1);
	double t_3 = 1.0 + t_1;
	double t_4 = x * t_1;
	double tmp;
	if ((x * x) <= 2e+45) {
		tmp = ((1.0 + ((x * t_3) * ((t_3 * t_0) * (t_3 * ((x * x) * t_2))))) / ((1.0 + (t_3 * ((x * x) * ((x * x) * t_3)))) * (1.0 + t_2))) / Math.E;
	} else if ((x * x) <= 5e+102) {
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_4 * (t_4 - x))))) / Math.E;
	} else {
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (x * x) * 0.5
	t_2 = (x * x) * (-1.0 - t_1)
	t_3 = 1.0 + t_1
	t_4 = x * t_1
	tmp = 0
	if (x * x) <= 2e+45:
		tmp = ((1.0 + ((x * t_3) * ((t_3 * t_0) * (t_3 * ((x * x) * t_2))))) / ((1.0 + (t_3 * ((x * x) * ((x * x) * t_3)))) * (1.0 + t_2))) / math.e
	elif (x * x) <= 5e+102:
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_4 * (t_4 - x))))) / math.e
	else:
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * 0.5)
	t_2 = Float64(Float64(x * x) * Float64(-1.0 - t_1))
	t_3 = Float64(1.0 + t_1)
	t_4 = Float64(x * t_1)
	tmp = 0.0
	if (Float64(x * x) <= 2e+45)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x * t_3) * Float64(Float64(t_3 * t_0) * Float64(t_3 * Float64(Float64(x * x) * t_2))))) / Float64(Float64(1.0 + Float64(t_3 * Float64(Float64(x * x) * Float64(Float64(x * x) * t_3)))) * Float64(1.0 + t_2))) / exp(1));
	elseif (Float64(x * x) <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * Float64(t_0 + Float64(t_0 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) * 0.125)))) / Float64(Float64(x * x) + Float64(t_4 * Float64(t_4 - x))))) / exp(1));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (x * x) * 0.5;
	t_2 = (x * x) * (-1.0 - t_1);
	t_3 = 1.0 + t_1;
	t_4 = x * t_1;
	tmp = 0.0;
	if ((x * x) <= 2e+45)
		tmp = ((1.0 + ((x * t_3) * ((t_3 * t_0) * (t_3 * ((x * x) * t_2))))) / ((1.0 + (t_3 * ((x * x) * ((x * x) * t_3)))) * (1.0 + t_2))) / 2.71828182845904523536;
	elseif ((x * x) <= 5e+102)
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_4 * (t_4 - x))))) / 2.71828182845904523536;
	else
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	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] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e+45], N[(N[(N[(1.0 + N[(N[(x * t$95$3), $MachinePrecision] * N[(N[(t$95$3 * t$95$0), $MachinePrecision] * N[(t$95$3 * N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$3 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+102], N[(N[(1.0 + N[(N[(x * N[(t$95$0 + N[(t$95$0 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(t$95$4 * N[(t$95$4 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.9999999999999999e45

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 93.6%

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

   9. Simplified 93.6%

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

   11. Applied egg-rr 94.9%

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

   if 1.9999999999999999e45 < (*.f64 x x) < 5e102

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 4.7%

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

   9. Simplified 4.7%

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

       1. distribute-lft-in N/A

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

       2. flip3-+ N/A

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

       3. associate-*r/ N/A

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

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

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

   11. Applied egg-rr 73.9%

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

   if 5e102 < (*.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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{1 + \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right) \cdot \left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-1 - \left(x \cdot x\right) \cdot 0.5\right)\right)\right)\right)\right)}{\left(1 + \left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-1 - \left(x \cdot x\right) \cdot 0.5\right)\right)}}{e}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{x \cdot \left(x \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.125\right)\right)}{x \cdot x + \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) - x\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ t_2 := t\_0 \cdot t\_1\\ t_3 := x \cdot t\_2\\ t_4 := \left(x \cdot x\right) \cdot 0.5\\ t_5 := 1 - x \cdot x\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + t\_4\right)\right)}{e}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 - \left(x \cdot x\right) \cdot \left(t\_2 \cdot t\_2\right)}{1 + t\_3}}{\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot t\_5}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - t\_3}{t\_5}}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot t\_4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* x t_0))
        (t_2 (* t_0 t_1))
        (t_3 (* x t_2))
        (t_4 (* (* x x) 0.5))
        (t_5 (- 1.0 (* x x))))
   (if (<= (* x x) 2e+37)
     (/ (+ 1.0 (* x (* x (+ 1.0 t_4)))) E)
     (if (<= (* x x) 2e+62)
       (/
        (/ (- 1.0 (* (* x x) (* t_2 t_2))) (+ 1.0 t_3))
        (* (+ 1.0 (* (* x x) (* x x))) t_5))
       (if (<= (* x x) 5e+153)
         (/ (/ (- 1.0 t_3) t_5) (+ 1.0 t_1))
         (* x (* x t_4)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = t_0 * t_1;
	double t_3 = x * t_2;
	double t_4 = (x * x) * 0.5;
	double t_5 = 1.0 - (x * x);
	double tmp;
	if ((x * x) <= 2e+37) {
		tmp = (1.0 + (x * (x * (1.0 + t_4)))) / ((double) M_E);
	} else if ((x * x) <= 2e+62) {
		tmp = ((1.0 - ((x * x) * (t_2 * t_2))) / (1.0 + t_3)) / ((1.0 + ((x * x) * (x * x))) * t_5);
	} else if ((x * x) <= 5e+153) {
		tmp = ((1.0 - t_3) / t_5) / (1.0 + t_1);
	} else {
		tmp = x * (x * t_4);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = t_0 * t_1;
	double t_3 = x * t_2;
	double t_4 = (x * x) * 0.5;
	double t_5 = 1.0 - (x * x);
	double tmp;
	if ((x * x) <= 2e+37) {
		tmp = (1.0 + (x * (x * (1.0 + t_4)))) / Math.E;
	} else if ((x * x) <= 2e+62) {
		tmp = ((1.0 - ((x * x) * (t_2 * t_2))) / (1.0 + t_3)) / ((1.0 + ((x * x) * (x * x))) * t_5);
	} else if ((x * x) <= 5e+153) {
		tmp = ((1.0 - t_3) / t_5) / (1.0 + t_1);
	} else {
		tmp = x * (x * t_4);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = x * t_0
	t_2 = t_0 * t_1
	t_3 = x * t_2
	t_4 = (x * x) * 0.5
	t_5 = 1.0 - (x * x)
	tmp = 0
	if (x * x) <= 2e+37:
		tmp = (1.0 + (x * (x * (1.0 + t_4)))) / math.e
	elif (x * x) <= 2e+62:
		tmp = ((1.0 - ((x * x) * (t_2 * t_2))) / (1.0 + t_3)) / ((1.0 + ((x * x) * (x * x))) * t_5)
	elif (x * x) <= 5e+153:
		tmp = ((1.0 - t_3) / t_5) / (1.0 + t_1)
	else:
		tmp = x * (x * t_4)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * t_0)
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(x * t_2)
	t_4 = Float64(Float64(x * x) * 0.5)
	t_5 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (Float64(x * x) <= 2e+37)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + t_4)))) / exp(1));
	elseif (Float64(x * x) <= 2e+62)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(t_2 * t_2))) / Float64(1.0 + t_3)) / Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(x * x))) * t_5));
	elseif (Float64(x * x) <= 5e+153)
		tmp = Float64(Float64(Float64(1.0 - t_3) / t_5) / Float64(1.0 + t_1));
	else
		tmp = Float64(x * Float64(x * t_4));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = x * t_0;
	t_2 = t_0 * t_1;
	t_3 = x * t_2;
	t_4 = (x * x) * 0.5;
	t_5 = 1.0 - (x * x);
	tmp = 0.0;
	if ((x * x) <= 2e+37)
		tmp = (1.0 + (x * (x * (1.0 + t_4)))) / 2.71828182845904523536;
	elseif ((x * x) <= 2e+62)
		tmp = ((1.0 - ((x * x) * (t_2 * t_2))) / (1.0 + t_3)) / ((1.0 + ((x * x) * (x * x))) * t_5);
	elseif ((x * x) <= 5e+153)
		tmp = ((1.0 - t_3) / t_5) / (1.0 + t_1);
	else
		tmp = x * (x * t_4);
	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 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e+37], N[(N[(1.0 + N[(x * N[(x * N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+62], N[(N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+153], N[(N[(N[(1.0 - t$95$3), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.99999999999999991e37

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 94.8%

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

   9. Simplified 94.8%

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

   if 1.99999999999999991e37 < (*.f64 x x) < 2.00000000000000007e62

   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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 3.6%

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

   10. Simplified 3.6%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{1 - x \cdot x} \]

       4. flip-- N/A

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

       5. frac-times N/A

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

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

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

   12. Applied egg-rr 3.6%

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

       1. *-rgt-identity N/A

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

       2. flip-- N/A

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

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

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

   14. Applied egg-rr 100.0%

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

   if 2.00000000000000007e62 < (*.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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 4.1%

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

   10. Simplified 4.1%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{1 - x \cdot x} \]

       4. flip-- N/A

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

       5. frac-times N/A

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

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

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

   12. Applied egg-rr 38.9%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

   14. Applied egg-rr 100.0%

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

   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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}{e}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{1 - \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{1 + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}{\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(1 - x \cdot x\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 - x \cdot x}}{1 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + t\_1\right)\right)}{e}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{x \cdot \left(t\_0 + t\_0 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.125\right)\right)}{x \cdot x + t\_2 \cdot \left(t\_2 - x\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* (* x x) 0.5)) (t_2 (* x t_1)))
   (if (<= (* x x) 5e+24)
     (/ (+ 1.0 (* x (* x (+ 1.0 t_1)))) E)
     (if (<= (* x x) 5e+102)
       (/
        (+
         1.0
         (/
          (* x (+ t_0 (* t_0 (* (* (* x x) (* (* x x) (* x x))) 0.125))))
          (+ (* x x) (* t_2 (- t_2 x)))))
        E)
       (+
        1.0
        (*
         (* x x)
         (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.16666666666666666))))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * 0.5;
	double t_2 = x * t_1;
	double tmp;
	if ((x * x) <= 5e+24) {
		tmp = (1.0 + (x * (x * (1.0 + t_1)))) / ((double) M_E);
	} else if ((x * x) <= 5e+102) {
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_2 * (t_2 - x))))) / ((double) M_E);
	} else {
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * 0.5;
	double t_2 = x * t_1;
	double tmp;
	if ((x * x) <= 5e+24) {
		tmp = (1.0 + (x * (x * (1.0 + t_1)))) / Math.E;
	} else if ((x * x) <= 5e+102) {
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_2 * (t_2 - x))))) / Math.E;
	} else {
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = (x * x) * 0.5
	t_2 = x * t_1
	tmp = 0
	if (x * x) <= 5e+24:
		tmp = (1.0 + (x * (x * (1.0 + t_1)))) / math.e
	elif (x * x) <= 5e+102:
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_2 * (t_2 - x))))) / math.e
	else:
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * 0.5)
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (Float64(x * x) <= 5e+24)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + t_1)))) / exp(1));
	elseif (Float64(x * x) <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * Float64(t_0 + Float64(t_0 * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) * 0.125)))) / Float64(Float64(x * x) + Float64(t_2 * Float64(t_2 - x))))) / exp(1));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = (x * x) * 0.5;
	t_2 = x * t_1;
	tmp = 0.0;
	if ((x * x) <= 5e+24)
		tmp = (1.0 + (x * (x * (1.0 + t_1)))) / 2.71828182845904523536;
	elseif ((x * x) <= 5e+102)
		tmp = (1.0 + ((x * (t_0 + (t_0 * (((x * x) * ((x * x) * (x * x))) * 0.125)))) / ((x * x) + (t_2 * (t_2 - x))))) / 2.71828182845904523536;
	else
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	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] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+24], N[(N[(1.0 + N[(x * N[(x * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+102], N[(N[(1.0 + N[(N[(x * N[(t$95$0 + N[(t$95$0 * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(t$95$2 * N[(t$95$2 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.00000000000000045e24

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 96.7%

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

   9. Simplified 96.7%

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

   if 5.00000000000000045e24 < (*.f64 x x) < 5e102

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 4.4%

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

   9. Simplified 4.4%

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

       1. distribute-lft-in N/A

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

       2. flip3-+ N/A

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

       3. associate-*r/ N/A

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

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

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

   11. Applied egg-rr 52.0%

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

   if 5e102 < (*.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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}{e}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{x \cdot \left(x \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.125\right)\right)}{x \cdot x + \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) - x\right)}}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ t_2 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + t\_2\right)\right)}{e}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - x \cdot \left(t\_0 \cdot t\_1\right)}{1 - x \cdot x}}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* x t_0)) (t_2 (* (* x x) 0.5)))
   (if (<= (* x x) 2e+62)
     (/ (+ 1.0 (* x (* x (+ 1.0 t_2)))) E)
     (if (<= (* x x) 5e+153)
       (/ (/ (- 1.0 (* x (* t_0 t_1))) (- 1.0 (* x x))) (+ 1.0 t_1))
       (* x (* x t_2))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = (x * x) * 0.5;
	double tmp;
	if ((x * x) <= 2e+62) {
		tmp = (1.0 + (x * (x * (1.0 + t_2)))) / ((double) M_E);
	} else if ((x * x) <= 5e+153) {
		tmp = ((1.0 - (x * (t_0 * t_1))) / (1.0 - (x * x))) / (1.0 + t_1);
	} else {
		tmp = x * (x * t_2);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = (x * x) * 0.5;
	double tmp;
	if ((x * x) <= 2e+62) {
		tmp = (1.0 + (x * (x * (1.0 + t_2)))) / Math.E;
	} else if ((x * x) <= 5e+153) {
		tmp = ((1.0 - (x * (t_0 * t_1))) / (1.0 - (x * x))) / (1.0 + t_1);
	} else {
		tmp = x * (x * t_2);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = x * t_0
	t_2 = (x * x) * 0.5
	tmp = 0
	if (x * x) <= 2e+62:
		tmp = (1.0 + (x * (x * (1.0 + t_2)))) / math.e
	elif (x * x) <= 5e+153:
		tmp = ((1.0 - (x * (t_0 * t_1))) / (1.0 - (x * x))) / (1.0 + t_1)
	else:
		tmp = x * (x * t_2)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * t_0)
	t_2 = Float64(Float64(x * x) * 0.5)
	tmp = 0.0
	if (Float64(x * x) <= 2e+62)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + t_2)))) / exp(1));
	elseif (Float64(x * x) <= 5e+153)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * Float64(t_0 * t_1))) / Float64(1.0 - Float64(x * x))) / Float64(1.0 + t_1));
	else
		tmp = Float64(x * Float64(x * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = x * t_0;
	t_2 = (x * x) * 0.5;
	tmp = 0.0;
	if ((x * x) <= 2e+62)
		tmp = (1.0 + (x * (x * (1.0 + t_2)))) / 2.71828182845904523536;
	elseif ((x * x) <= 5e+153)
		tmp = ((1.0 - (x * (t_0 * t_1))) / (1.0 - (x * x))) / (1.0 + t_1);
	else
		tmp = x * (x * t_2);
	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 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e+62], N[(N[(1.0 + N[(x * N[(x * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+153], N[(N[(N[(1.0 - N[(x * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.00000000000000007e62

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 91.3%

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

   9. Simplified 91.3%

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

   if 2.00000000000000007e62 < (*.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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 4.1%

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

   10. Simplified 4.1%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{1 - x \cdot x} \]

       4. flip-- N/A

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

       5. frac-times N/A

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

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

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

   12. Applied egg-rr 38.9%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

   14. Applied egg-rr 100.0%

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

   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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}{e}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{1 - x \cdot x}}{1 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-8)
   (/ (+ 1.0 (* x (* x (+ 1.0 (* (* x x) 0.5))))) E)
   (+
    1.0
    (* (* x x) (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.16666666666666666))))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) * 0.5))))) / ((double) M_E);
	} else {
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) * 0.5))))) / Math.E;
	} else {
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) * 0.5))))) / math.e
	else:
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(Float64(x * x) * 0.5))))) / exp(1));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 5e-8)
		tmp = (1.0 + (x * (x * (1.0 + ((x * x) * 0.5))))) / 2.71828182845904523536;
	else
		tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], N[(N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 83.2%

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

   10. Simplified 83.2%

       \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 88.0% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-8)
   (/ (/ 1.0 E) (/ 1.0 (+ (* x x) 1.0)))
   (+ 1.0 (* x (* x (+ 1.0 (* (* x x) 0.5)))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = (1.0 / ((double) M_E)) / (1.0 / ((x * x) + 1.0));
	} else {
		tmp = 1.0 + (x * (x * (1.0 + ((x * x) * 0.5))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = (1.0 / Math.E) / (1.0 / ((x * x) + 1.0));
	} else {
		tmp = 1.0 + (x * (x * (1.0 + ((x * x) * 0.5))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = (1.0 / math.e) / (1.0 / ((x * x) + 1.0))
	else:
		tmp = 1.0 + (x * (x * (1.0 + ((x * x) * 0.5))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		tmp = Float64(Float64(1.0 / exp(1)) / Float64(1.0 / Float64(Float64(x * x) + 1.0)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * 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) <= 5e-8)
		tmp = (1.0 / 2.71828182845904523536) / (1.0 / ((x * x) + 1.0));
	else
		tmp = 1.0 + (x * (x * (1.0 + ((x * x) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], N[(N[(1.0 / E), $MachinePrecision] / N[(1.0 / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{1 + x \cdot x}}{e} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. flip-+ N/A

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

       4. clear-num N/A

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

       5. un-div-inv N/A

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

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

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

       7. /-lowering-/.f64 N/A

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

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

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

       9. clear-num N/A

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

       10. flip-+ N/A

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

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

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

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

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

       13. *-lowering-*.f64 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 4.9999999999999998e-8 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.2%

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

   10. Simplified 74.2%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

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

Alternative 9: 88.0% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-8)
   (/ (/ 1.0 E) (/ 1.0 (+ (* x x) 1.0)))
   (* x (* x (+ 1.0 (* (* x x) 0.5))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = (1.0 / ((double) M_E)) / (1.0 / ((x * x) + 1.0));
	} else {
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = (1.0 / Math.E) / (1.0 / ((x * x) + 1.0));
	} else {
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = (1.0 / math.e) / (1.0 / ((x * x) + 1.0))
	else:
		tmp = x * (x * (1.0 + ((x * x) * 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		tmp = Float64(Float64(1.0 / exp(1)) / Float64(1.0 / Float64(Float64(x * x) + 1.0)));
	else
		tmp = Float64(x * Float64(x * 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) <= 5e-8)
		tmp = (1.0 / 2.71828182845904523536) / (1.0 / ((x * x) + 1.0));
	else
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], N[(N[(1.0 / E), $MachinePrecision] / N[(1.0 / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{1 + x \cdot x}}{e} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. flip-+ N/A

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

       4. clear-num N/A

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

       5. un-div-inv N/A

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

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

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

       7. /-lowering-/.f64 N/A

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

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

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

       9. clear-num N/A

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

       10. flip-+ N/A

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

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

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

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

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

       13. *-lowering-*.f64 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 4.9999999999999998e-8 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. cube-mult N/A

          \[\leadsto \left(x \cdot {x}^{3}\right) \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right)} \]

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. lft-mult-inverse N/A

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

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

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 74.2%

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

   13. Simplified 74.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

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

Alternative 10: 88.0% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-8)
   (/ (+ (* x x) 1.0) E)
   (* x (* x (+ 1.0 (* (* x x) 0.5))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = ((x * x) + 1.0) / ((double) M_E);
	} else {
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = ((x * x) + 1.0) / Math.E;
	} else {
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = ((x * x) + 1.0) / math.e
	else:
		tmp = x * (x * (1.0 + ((x * x) * 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	else
		tmp = Float64(x * Float64(x * 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) <= 5e-8)
		tmp = ((x * x) + 1.0) / 2.71828182845904523536;
	else
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. cube-mult N/A

          \[\leadsto \left(x \cdot {x}^{3}\right) \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)\right)} \]

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. lft-mult-inverse N/A

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

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

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 74.2%

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

   13. Simplified 74.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

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

Alternative 11: 88.0% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-8) (/ (+ (* x x) 1.0) E) (* x (* x (* (* x x) 0.5)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = ((x * x) + 1.0) / ((double) M_E);
	} else {
		tmp = x * (x * ((x * x) * 0.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 5e-8) {
		tmp = ((x * x) + 1.0) / Math.E;
	} else {
		tmp = x * (x * ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-8:
		tmp = ((x * x) + 1.0) / math.e
	else:
		tmp = x * (x * ((x * x) * 0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-8)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 5e-8)
		tmp = ((x * x) + 1.0) / 2.71828182845904523536;
	else
		tmp = x * (x * ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-8], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e-8

   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. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

      4. exp-diff N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

      8. metadata-eval N/A

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

      9. exp-1-e N/A

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

      10. E-lowering-E.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 74.2%

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

   13. Simplified 74.2%

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

4. Final simplification 88.2%

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

Alternative 12: 88.3% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 + N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $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. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

   4. exp-diff N/A

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

   8. metadata-eval N/A

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

   9. exp-1-e N/A

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

   10. E-lowering-E.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 88.2%

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

9. Simplified 88.2%

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

Alternative 13: 75.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.78:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.78) (/ 1.0 E) (+ (* x x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.78) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x * x) + 1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.78) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x * x) + 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.78:
		tmp = 1.0 / math.e
	else:
		tmp = (x * x) + 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.78)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(Float64(x * x) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.78)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x * x) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.78], N[(1.0 / E), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.78000000000000003

   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. exp-lowering-exp.f64 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \color{blue}{e^{-1}} \]
   8. 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. e-exp-1 N/A

         \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\mathsf{E}\left(\right)}\right) \]

      5. E-lowering-E.f64 98.9%

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

   9. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 0.78000000000000003 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 50.8%

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

   10. Simplified 50.8%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.78:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.37:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.37) (/ 1.0 E) (* x x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.37) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.37) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.37:
		tmp = 1.0 / math.e
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.37)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.37)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.37], N[(1.0 / E), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.37

   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. exp-lowering-exp.f64 98.9%

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

   7. Simplified 98.9%

      \[\leadsto \color{blue}{e^{-1}} \]
   8. 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. e-exp-1 N/A

         \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\mathsf{E}\left(\right)}\right) \]

      5. E-lowering-E.f64 98.9%

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

   9. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 0.37 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 50.8%

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

   10. Simplified 50.8%

       \[\leadsto \color{blue}{1 + x \cdot x} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \color{blue}{x} \]

       2. *-lowering-*.f64 50.8%

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

   13. Simplified 50.8%

       \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 35.2% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.2) 1.0 (* x x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.2d0) then
        tmp = 1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.2:
		tmp = 1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.2)
		tmp = 1.0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.2)
		tmp = 1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.2], 1.0, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.20000000000000001

   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 17.8%

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

   7. Simplified 17.8%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 17.8%

       \[\leadsto \color{blue}{1} \]

   if 0.20000000000000001 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 50.8%

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

   10. Simplified 50.8%

       \[\leadsto \color{blue}{1 + x \cdot x} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto x \cdot \color{blue}{x} \]

       2. *-lowering-*.f64 50.8%

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

   13. Simplified 50.8%

       \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.0% 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. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto e^{x \cdot x + \left(\mathsf{neg}\left(-1 \cdot -1\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto e^{x \cdot x - -1 \cdot -1} \]

   4. exp-diff N/A

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot x\right)\right), \left(e^{\color{blue}{-1 \cdot -1}}\right)\right) \]

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

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

   8. metadata-eval N/A

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

   9. exp-1-e N/A

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

   10. E-lowering-E.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 77.5%

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

9. Simplified 77.5%

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

10. Final simplification 77.5%

    \[\leadsto \frac{x \cdot x + 1}{e} \]
11. Add Preprocessing

Alternative 17: 10.5% accurate, 106.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 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 55.0%

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

7. Simplified 55.0%

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

8. Taylor expanded in x around 0

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

10. Simplified 11.1%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))