exp neg sub

Percentage Accurate: 100.0% → 99.9%

Time: 4.8s

Alternatives: 9

Speedup: 1.0×

• 50.3% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(e^{3 \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (exp (* 3.0 (fma x x -1.0))) 0.3333333333333333))

C

double code(double x) {
	return pow(exp((3.0 * fma(x, x, -1.0))), 0.3333333333333333);
}

Julia

function code(x)
	return exp(Float64(3.0 * fma(x, x, -1.0))) ^ 0.3333333333333333
end

Wolfram

code[x_] := N[Power[N[Exp[N[(3.0 * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{x \cdot x + -1} \cdot e^{x \cdot x + -1}\right) \cdot e^{x \cdot x + -1}}} \]

   2. pow1/3 99.2%

      \[\leadsto \color{blue}{{\left(\left(e^{x \cdot x + -1} \cdot e^{x \cdot x + -1}\right) \cdot e^{x \cdot x + -1}\right)}^{0.3333333333333333}} \]

   3. add-exp-log 100.0%

      \[\leadsto {\color{blue}{\left(e^{\log \left(\left(e^{x \cdot x + -1} \cdot e^{x \cdot x + -1}\right) \cdot e^{x \cdot x + -1}\right)}\right)}}^{0.3333333333333333} \]

   4. pow3 100.0%

      \[\leadsto {\left(e^{\log \color{blue}{\left({\left(e^{x \cdot x + -1}\right)}^{3}\right)}}\right)}^{0.3333333333333333} \]

   5. log-pow 100.0%

      \[\leadsto {\left(e^{\color{blue}{3 \cdot \log \left(e^{x \cdot x + -1}\right)}}\right)}^{0.3333333333333333} \]

   6. add-log-exp 100.0%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(x \cdot x + -1\right)}}\right)}^{0.3333333333333333} \]

   7. fma-define 100.0%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{0.3333333333333333} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(e^{3 \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.3333333333333333}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.97:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.97) (/ 1.0 E) (exp x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.97) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.97) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.97:
		tmp = 1.0 / math.e
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.97)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.97)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.97], N[(1.0 / E), $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.96999999999999997

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

         \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

      2. sqr-neg 100.0%

         \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

      3. associate--r- 100.0%

         \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

      1. difference-of-sqr--1 100.0%

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

      2. exp-prod 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 66.7%

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

      1. exp-1-e 66.7%

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

   9. Simplified 66.7%

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

   10. Taylor expanded in x around 0 67.9%

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

   if 0.96999999999999997 < x

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

         \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

      2. sqr-neg 99.9%

         \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

      3. associate--r- 99.9%

         \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

      1. difference-of-sqr--1 99.9%

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

      2. exp-prod 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. exp-1-e 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 98.6%

       \[\leadsto {e}^{\color{blue}{x}} \]
   11. Step-by-step derivation

       1. add-log-exp 98.6%

          \[\leadsto \color{blue}{\log \left(e^{{e}^{x}}\right)} \]

       2. *-un-lft-identity 98.6%

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

       3. log-prod 98.6%

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

       4. metadata-eval 98.6%

          \[\leadsto \color{blue}{0} + \log \left(e^{{e}^{x}}\right) \]

       5. add-log-exp 98.6%

          \[\leadsto 0 + \color{blue}{{e}^{x}} \]

       6. e-exp-1 98.6%

          \[\leadsto 0 + {\color{blue}{\left(e^{1}\right)}}^{x} \]

       7. pow-exp 98.6%

          \[\leadsto 0 + \color{blue}{e^{1 \cdot x}} \]

       8. *-un-lft-identity 98.6%

          \[\leadsto 0 + e^{\color{blue}{x}} \]

   12. Applied egg-rr 98.6%

       \[\leadsto \color{blue}{0 + e^{x}} \]
   13. Step-by-step derivation

       1. +-lft-identity 98.6%

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

   14. Simplified 98.6%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

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(-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]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 5: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

Alternative 6: 63.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.95) (/ 1.0 E) (+ 1.0 (* x (+ 1.0 (* x 0.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.95) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.95) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.95:
		tmp = 1.0 / math.e
	else:
		tmp = 1.0 + (x * (1.0 + (x * 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.95], N[(1.0 / E), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

         \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

      2. sqr-neg 100.0%

         \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

      3. associate--r- 100.0%

         \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

      1. difference-of-sqr--1 100.0%

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

      2. exp-prod 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 66.7%

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

      1. exp-1-e 66.7%

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

   9. Simplified 66.7%

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

   10. Taylor expanded in x around 0 67.9%

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

   if 0.94999999999999996 < x

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

         \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

      2. sqr-neg 99.9%

         \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

      3. associate--r- 99.9%

         \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

      1. difference-of-sqr--1 99.9%

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

      2. exp-prod 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. exp-1-e 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf 98.6%

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

   11. Taylor expanded in x around 0 52.6%

       \[\leadsto \color{blue}{1 + x \cdot \left(\log e + 0.5 \cdot \left(x \cdot {\log e}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. log-E 52.6%

          \[\leadsto 1 + x \cdot \left(\color{blue}{1} + 0.5 \cdot \left(x \cdot {\log e}^{2}\right)\right) \]

       2. log-E 52.6%

          \[\leadsto 1 + x \cdot \left(1 + 0.5 \cdot \left(x \cdot {\color{blue}{1}}^{2}\right)\right) \]

       3. metadata-eval 52.6%

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

   13. Simplified 52.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e} - \frac{-1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ x E) (/ -1.0 E)))

C

double code(double x) {
	return (x / ((double) M_E)) - (-1.0 / ((double) M_E));
}

Java

public static double code(double x) {
	return (x / Math.E) - (-1.0 / Math.E);
}

Python

def code(x):
	return (x / math.e) - (-1.0 / math.e)

Julia

function code(x)
	return Float64(Float64(x / exp(1)) - Float64(-1.0 / exp(1)))
end

MATLAB

function tmp = code(x)
	tmp = (x / 2.71828182845904523536) - (-1.0 / 2.71828182845904523536);
end

Wolfram

code[x_] := N[(N[(x / E), $MachinePrecision] - N[(-1.0 / E), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in x around 0 50.6%

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

    1. log-E 50.6%

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

    2. metadata-eval 50.6%

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

    3. log-E 50.6%

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

    4. associate-/l* 50.6%

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

    5. log-E 50.6%

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

    6. metadata-eval 50.6%

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

    7. /-rgt-identity 50.6%

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

    8. associate-*r/ 50.6%

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

    9. associate-*l/ 50.6%

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

    10. /-rgt-identity 50.6%

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

12. Simplified 50.6%

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

    1. +-commutative 50.6%

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

    2. fma-define 50.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{e}, \frac{1}{e}\right)} \]

    3. frac-2neg 50.6%

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{e}, \color{blue}{\frac{-1}{-e}}\right) \]

    4. metadata-eval 50.6%

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{e}, \frac{\color{blue}{-1}}{-e}\right) \]

    5. distribute-frac-neg2 50.6%

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{e}, \color{blue}{-\frac{-1}{e}}\right) \]

    6. fmm-undef 50.6%

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

    7. un-div-inv 50.6%

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

14. Applied egg-rr 50.6%

    \[\leadsto \color{blue}{\frac{x}{e} - \frac{-1}{e}} \]
15. Add Preprocessing

Alternative 8: 49.8% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \frac{x + 1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x 1.0) E))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in x around 0 50.6%

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

    1. log-E 50.6%

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

    2. metadata-eval 50.6%

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

    3. log-E 50.6%

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

    4. associate-/l* 50.6%

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

    5. log-E 50.6%

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

    6. metadata-eval 50.6%

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

    7. /-rgt-identity 50.6%

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

    8. associate-*r/ 50.6%

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

    9. associate-*l/ 50.6%

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

    10. /-rgt-identity 50.6%

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

    11. distribute-rgt1-in 50.6%

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

    12. associate-*r/ 50.6%

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

    13. *-rgt-identity 50.6%

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

    14. +-commutative 50.6%

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

12. Simplified 50.6%

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

13. Final simplification 50.6%

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

Alternative 9: 50.5% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 E))

C

double code(double x) {
	return 1.0 / ((double) M_E);
}

Java

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

Python

def code(x):
	return 1.0 / math.e

Julia

function code(x)
	return Float64(1.0 / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(1.0 / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   2. sqr-neg 100.0%

      \[\leadsto e^{0 - \left(1 - \color{blue}{\left(-x\right) \cdot \left(-x\right)}\right)} \]

   3. associate--r- 100.0%

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 74.9%

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

   1. exp-1-e 74.9%

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

9. Simplified 74.9%

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

10. Taylor expanded in x around 0 51.2%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))