exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 6

Speedup: 1.0×

• 50.2% 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, 0.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (pow (exp x) x) E))

C

double code(double x) {
	return pow(exp(x), x) / ((double) M_E);
}

Java

public static double code(double x) {
	return Math.pow(Math.exp(x), x) / Math.E;
}

Python

def code(x):
	return math.pow(math.exp(x), x) / math.e

Julia

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

MATLAB

function tmp = code(x)
	tmp = (exp(x) ^ x) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   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. 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%

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

5. Applied egg-rr 100.0%

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

   1. exp-sum 100.0%

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

   2. unpow-prod-down 72.6%

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

   3. exp-prod 72.6%

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

   4. *-un-lft-identity 72.6%

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

7. Applied egg-rr 72.6%

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

   1. metadata-eval 72.6%

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

   2. sub-neg 72.6%

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

   3. exp-diff 72.6%

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

   4. associate-*r/ 72.6%

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

   5. pow-plus 100.0%

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

   6. +-commutative 100.0%

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

   7. exp-1-e 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(e^{-2 + x \cdot 2}\right)}^{\left(0.5 + x \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (exp (+ -2.0 (* x 2.0))) (+ 0.5 (* x 0.5))))

C

double code(double x) {
	return pow(exp((-2.0 + (x * 2.0))), (0.5 + (x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(((-2.0d0) + (x * 2.0d0))) ** (0.5d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return Math.pow(Math.exp((-2.0 + (x * 2.0))), (0.5 + (x * 0.5)));
}

Python

def code(x):
	return math.pow(math.exp((-2.0 + (x * 2.0))), (0.5 + (x * 0.5)))

Julia

function code(x)
	return exp(Float64(-2.0 + Float64(x * 2.0))) ^ Float64(0.5 + Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = exp((-2.0 + (x * 2.0))) ^ (0.5 + (x * 0.5));
end

Wolfram

code[x_] := N[Power[N[Exp[N[(-2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   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. 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%

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

   5. unpow-prod-up 76.9%

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

5. Applied egg-rr 76.9%

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

   1. unpow-1 76.9%

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

7. Simplified 76.9%

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

   1. inv-pow 76.9%

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

   2. unpow-prod-up 100.0%

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

   3. pow-exp 99.9%

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

   4. add-log-exp 99.9%

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

   5. log-pow 99.9%

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

   6. sqr-pow 99.9%

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

   7. add-exp-log 100.0%

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

   8. pow-prod-down 100.0%

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

   9. pow2 100.0%

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

   10. div-inv 100.0%

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

   11. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. exp-prod 100.0%

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

    2. *-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. distribute-rgt-in 100.0%

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

    5. metadata-eval 100.0%

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

    6. *-commutative 100.0%

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

    7. +-commutative 100.0%

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

    8. distribute-lft-in 100.0%

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

    9. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

    \[\leadsto {\left(e^{-2 + x \cdot 2}\right)}^{\left(0.5 + x \cdot 0.5\right)} \]

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return exp(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[Power[N[Exp[N[(x + -1.0), $MachinePrecision]], $MachinePrecision], N[(x + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   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. 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. *-commutative 99.9%

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

   3. exp-prod 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return exp(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[Power[N[Exp[N[(x + 1.0), $MachinePrecision]], $MachinePrecision], N[(x + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   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. 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%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 5: 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 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   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. Final simplification 99.9%

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

Alternative 6: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Exp[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sqr-neg 99.9%

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

   2. neg-sub0 99.9%

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

   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. Taylor expanded in x around 0 49.3%

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

5. Final simplification 49.3%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))