exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 14

Speedup: 1.0×

• 49.6% 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

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

FPCore

(FPCore (x) :precision binary64 (pow E (+ (* x x) -1.0)))

C

double code(double x) {
	return pow(((double) M_E), ((x * x) + -1.0));
}

Java

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

Python

def code(x):
	return math.pow(math.e, ((x * x) + -1.0))

Julia

function code(x)
	return exp(1) ^ Float64(Float64(x * x) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = 2.71828182845904523536 ^ ((x * x) + -1.0);
end

Wolfram

code[x_] := N[Power[E, 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. Step-by-step derivation

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. difference-of-sqr--1 N/A

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

   8. difference-of-sqr-1 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. exp-prod N/A

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

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

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

   14. metadata-eval N/A

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

   15. exp-1-e N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. difference-of-sqr-1 N/A

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

   21. difference-of-sqr--1 N/A

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{e}^{\left(x \cdot x + -1\right)}} \]
7. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-10}:\\ \;\;\;\;\frac{x \cdot x + 1}{e}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-10) (/ (+ (* x x) 1.0) E) (exp (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / ((double) M_E);
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / Math.E;
	} else {
		tmp = Math.exp((x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 1e-10:
		tmp = ((x * x) + 1.0) / math.e
	else:
		tmp = math.exp((x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1e-10)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 1e-10)
		tmp = ((x * x) + 1.0) / 2.71828182845904523536;
	else
		tmp = exp((x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-10], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x)

   1. Initial program 100.0%

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

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

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.7%

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

Alternative 3: 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 4: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\\ t_2 := x \cdot \left(x \cdot t\_1\right)\\ t_3 := x \cdot t\_0\\ t_4 := \left(1 - x \cdot x\right) \cdot \left(1 + t\_3\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-10}:\\ \;\;\;\;\frac{x \cdot x + 1}{e}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{1 - \left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_2\right)}{1 + t\_2}}{t\_4}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot t\_3\right)\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* x (* (* x x) t_0)))
        (t_2 (* x (* x t_1)))
        (t_3 (* x t_0))
        (t_4 (* (- 1.0 (* x x)) (+ 1.0 t_3))))
   (if (<= (* x x) 1e-10)
     (/ (+ (* x x) 1.0) E)
     (if (<= (* x x) 2e+72)
       (/ (/ (- 1.0 (* (* x x) (* t_1 t_2))) (+ 1.0 t_2)) t_4)
       (if (<= (* x x) 2e+102)
         (/ (- 1.0 (* (* x x) (* x (* x t_3)))) t_4)
         (+
          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 * x) * t_0);
	double t_2 = x * (x * t_1);
	double t_3 = x * t_0;
	double t_4 = (1.0 - (x * x)) * (1.0 + t_3);
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / ((double) M_E);
	} else if ((x * x) <= 2e+72) {
		tmp = ((1.0 - ((x * x) * (t_1 * t_2))) / (1.0 + t_2)) / t_4;
	} else if ((x * x) <= 2e+102) {
		tmp = (1.0 - ((x * x) * (x * (x * t_3)))) / t_4;
	} 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 * x) * t_0);
	double t_2 = x * (x * t_1);
	double t_3 = x * t_0;
	double t_4 = (1.0 - (x * x)) * (1.0 + t_3);
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / Math.E;
	} else if ((x * x) <= 2e+72) {
		tmp = ((1.0 - ((x * x) * (t_1 * t_2))) / (1.0 + t_2)) / t_4;
	} else if ((x * x) <= 2e+102) {
		tmp = (1.0 - ((x * x) * (x * (x * t_3)))) / t_4;
	} 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 * x) * t_0)
	t_2 = x * (x * t_1)
	t_3 = x * t_0
	t_4 = (1.0 - (x * x)) * (1.0 + t_3)
	tmp = 0
	if (x * x) <= 1e-10:
		tmp = ((x * x) + 1.0) / math.e
	elif (x * x) <= 2e+72:
		tmp = ((1.0 - ((x * x) * (t_1 * t_2))) / (1.0 + t_2)) / t_4
	elif (x * x) <= 2e+102:
		tmp = (1.0 - ((x * x) * (x * (x * t_3)))) / t_4
	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(x * Float64(Float64(x * x) * t_0))
	t_2 = Float64(x * Float64(x * t_1))
	t_3 = Float64(x * t_0)
	t_4 = Float64(Float64(1.0 - Float64(x * x)) * Float64(1.0 + t_3))
	tmp = 0.0
	if (Float64(x * x) <= 1e-10)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	elseif (Float64(x * x) <= 2e+72)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(t_1 * t_2))) / Float64(1.0 + t_2)) / t_4);
	elseif (Float64(x * x) <= 2e+102)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(x * Float64(x * t_3)))) / t_4);
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(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 * x) * t_0);
	t_2 = x * (x * t_1);
	t_3 = x * t_0;
	t_4 = (1.0 - (x * x)) * (1.0 + t_3);
	tmp = 0.0;
	if ((x * x) <= 1e-10)
		tmp = ((x * x) + 1.0) / 2.71828182845904523536;
	elseif ((x * x) <= 2e+72)
		tmp = ((1.0 - ((x * x) * (t_1 * t_2))) / (1.0 + t_2)) / t_4;
	elseif ((x * x) <= 2e+102)
		tmp = (1.0 - ((x * x) * (x * (x * t_3)))) / t_4;
	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[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-10], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+72], N[(N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+102], N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x) < 1.99999999999999989e72

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   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 95.2%

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

   7. Simplified 95.2%

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

   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 4.3%

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

   10. Simplified 4.3%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   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. metadata-eval N/A

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

       3. flip-- N/A

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

       4. associate-/l/ N/A

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

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

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

   12. Applied egg-rr 4.3%

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

       1. flip-- N/A

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

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

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

   14. Applied egg-rr 55.3%

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

   if 1.99999999999999989e72 < (*.f64 x x) < 1.99999999999999995e102

   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 3.9%

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

   10. Simplified 3.9%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   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. metadata-eval N/A

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

       3. flip-- N/A

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

       4. associate-/l/ N/A

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

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

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

   12. Applied egg-rr 100.0%

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

   if 1.99999999999999995e102 < (*.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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-10}:\\ \;\;\;\;\frac{x \cdot x + 1}{e}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}{1 + x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}}{\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-10}:\\ \;\;\;\;\frac{x \cdot x + 1}{e}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)}{\left(1 - x \cdot x\right) \cdot \left(1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= (* x x) 1e-10)
     (/ (+ (* x x) 1.0) E)
     (if (<= (* x x) 2e+102)
       (/ (- 1.0 (* (* x x) (* x (* x t_0)))) (* (- 1.0 (* x x)) (+ 1.0 t_0)))
       (+
        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 * x));
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / ((double) M_E);
	} else if ((x * x) <= 2e+102) {
		tmp = (1.0 - ((x * x) * (x * (x * t_0)))) / ((1.0 - (x * x)) * (1.0 + t_0));
	} 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 * x));
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / Math.E;
	} else if ((x * x) <= 2e+102) {
		tmp = (1.0 - ((x * x) * (x * (x * t_0)))) / ((1.0 - (x * x)) * (1.0 + t_0));
	} 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 * x))
	tmp = 0
	if (x * x) <= 1e-10:
		tmp = ((x * x) + 1.0) / math.e
	elif (x * x) <= 2e+102:
		tmp = (1.0 - ((x * x) * (x * (x * t_0)))) / ((1.0 - (x * x)) * (1.0 + t_0))
	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 * Float64(x * x)))
	tmp = 0.0
	if (Float64(x * x) <= 1e-10)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	elseif (Float64(x * x) <= 2e+102)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(x * Float64(x * t_0)))) / Float64(Float64(1.0 - Float64(x * x)) * Float64(1.0 + t_0)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(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 * x));
	tmp = 0.0;
	if ((x * x) <= 1e-10)
		tmp = ((x * x) + 1.0) / 2.71828182845904523536;
	elseif ((x * x) <= 2e+102)
		tmp = (1.0 - ((x * x) * (x * (x * t_0)))) / ((1.0 - (x * x)) * (1.0 + t_0));
	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 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-10], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+102], N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x) < 1.99999999999999995e102

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   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 96.6%

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

   7. Simplified 96.6%

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

   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 4.2%

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

   10. Simplified 4.2%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   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. metadata-eval N/A

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

       3. flip-- N/A

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

       4. associate-/l/ N/A

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

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

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

   12. Applied egg-rr 32.2%

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

   if 1.99999999999999995e102 < (*.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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 93.6%

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

Alternative 6: 91.3% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-10)
   (/ (+ (* x x) 1.0) 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) <= 1e-10) {
		tmp = ((x * x) + 1.0) / ((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) <= 1e-10) {
		tmp = ((x * x) + 1.0) / 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) <= 1e-10:
		tmp = ((x * x) + 1.0) / 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) <= 1e-10)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(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) <= 1e-10)
		tmp = ((x * x) + 1.0) / 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], 1e-10], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x)

   1. Initial program 100.0%

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

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

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 83.8%

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

   10. Simplified 83.8%

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

4. Final simplification 91.2%

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

Alternative 7: 87.8% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-10)
   (/ (+ (* x x) 1.0) E)
   (+ 1.0 (* x (* x (+ 1.0 (* (* x x) 0.5)))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-10) {
		tmp = ((x * x) + 1.0) / ((double) M_E);
	} 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) <= 1e-10) {
		tmp = ((x * x) + 1.0) / Math.E;
	} else {
		tmp = 1.0 + (x * (x * (1.0 + ((x * x) * 0.5))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 1e-10:
		tmp = ((x * x) + 1.0) / math.e
	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) <= 1e-10)
		tmp = Float64(Float64(Float64(x * x) + 1.0) / exp(1));
	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) <= 1e-10)
		tmp = ((x * x) + 1.0) / 2.71828182845904523536;
	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], 1e-10], N[(N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / E), $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) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x)

   1. Initial program 100.0%

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

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

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.6%

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

   10. Simplified 77.6%

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

4. Final simplification 87.8%

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

Alternative 8: 87.8% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-10)
   (/ (+ (* x x) 1.0) E)
   (* x (* x (+ 1.0 (* (* x x) 0.5))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-10) {
		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) <= 1e-10) {
		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) <= 1e-10:
		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) <= 1e-10)
		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) <= 1e-10)
		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], 1e-10], 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) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x)

   1. Initial program 100.0%

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

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

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.6%

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

   10. Simplified 77.6%

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

   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. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. lft-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-commutative N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 77.6%

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

   13. Simplified 77.6%

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

4. Final simplification 87.8%

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

Alternative 9: 87.8% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-10) (/ (+ (* x x) 1.0) E) (* x (* x (* (* x x) 0.5)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1e-10) {
		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) <= 1e-10) {
		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) <= 1e-10:
		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) <= 1e-10)
		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) <= 1e-10)
		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], 1e-10], 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) < 1.00000000000000004e-10

   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. flip-+ N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. difference-of-sqr--1 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. exp-prod N/A

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

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

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. difference-of-sqr-1 N/A

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

      21. difference-of-sqr--1 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. log-E N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. associate-/l* N/A

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

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

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

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

   9. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (*.f64 x x)

   1. Initial program 100.0%

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

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

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.6%

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

   10. Simplified 77.6%

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

   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 77.6%

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

   13. Simplified 77.6%

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

4. Final simplification 87.8%

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

Alternative 10: 76.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.79:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.79) (/ 1.0 E) (+ (* x x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.79) {
		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) <= 0.79) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x * x) + 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.79:
		tmp = 1.0 / math.e
	else:
		tmp = (x * x) + 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.79)
		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) <= 0.79)
		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], 0.79], 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) < 0.79000000000000004

   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 99.5%

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

   7. Simplified 99.5%

      \[\leadsto \color{blue}{e^{-1}} \]
   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 99.5%

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

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 0.79000000000000004 < (*.f64 x x)

   1. Initial program 100.0%

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

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

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

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

   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 53.1%

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

   10. Simplified 53.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.79:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% 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 99.5%

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

   7. Simplified 99.5%

      \[\leadsto \color{blue}{e^{-1}} \]
   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 99.5%

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

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{1}{e}} \]

   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 99.4%

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

   7. Simplified 99.4%

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

   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 53.1%

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

   10. Simplified 53.1%

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

   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 53.1%

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

   13. Simplified 53.1%

       \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 35.4% 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 99.4%

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

   7. Simplified 99.4%

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

   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 53.1%

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

   10. Simplified 53.1%

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

   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 53.1%

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

   13. Simplified 53.1%

       \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.4% 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. flip-+ N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. difference-of-sqr--1 N/A

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

   8. difference-of-sqr-1 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. exp-prod N/A

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

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

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

   14. metadata-eval N/A

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

   15. exp-1-e N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. difference-of-sqr-1 N/A

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

   21. difference-of-sqr--1 N/A

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

   1. log-E N/A

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

   2. metadata-eval N/A

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

   3. log-E N/A

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

   4. log-E N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. log-E N/A

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

   8. associate-/l* N/A

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

   9. log-E N/A

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

   10. metadata-eval N/A

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

   11. distribute-rgt1-in N/A

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

   12. *-commutative N/A

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

   13. associate-*l/ N/A

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

   14. *-lft-identity N/A

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

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

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

9. Simplified 74.5%

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

10. Final simplification 74.5%

    \[\leadsto \frac{x \cdot x + 1}{e} \]
11. Add Preprocessing

Alternative 14: 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 62.1%

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

7. Simplified 62.1%

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

8. Taylor expanded in x around 0

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

10. Simplified 9.9%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))