exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 5

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return exp(1) ^ fma(x, x, -1.0)
end

Wolfram

code[x_] := N[Power[E, N[(x * 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. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. exp-1-e 100.0%

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

   4. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 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 3: 75.7% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-10], N[(1.0 / E), $MachinePrecision], N[(x * N[(x / E), $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. 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. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. unpow2 100.0%

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

      3. fma-undefine 100.0%

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

      4. metadata-eval 100.0%

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

      5. rec-exp 100.0%

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

      6. e-exp-1 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. *-commutative 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 99.5%

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

   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. 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. Taylor expanded in x around 0 53.1%

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

      1. distribute-rgt1-in 53.1%

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

      2. unpow2 53.1%

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

      3. fma-undefine 53.1%

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

      4. metadata-eval 53.1%

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

      5. rec-exp 53.1%

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

      6. e-exp-1 53.1%

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

      7. associate-*r/ 53.1%

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

      8. metadata-eval 53.1%

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

      9. distribute-rgt-neg-in 53.1%

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

      10. *-commutative 53.1%

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

      11. neg-mul-1 53.1%

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

      12. remove-double-neg 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in x around inf 53.1%

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

      1. unpow2 53.1%

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

      2. associate-/l* 53.1%

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

   10. Applied egg-rr 53.1%

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

Alternative 4: 76.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1 + \left(x \cdot x + 2\right)}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ -1.0 (+ (* x x) 2.0)) E))

C

double code(double x) {
	return (-1.0 + ((x * x) + 2.0)) / ((double) M_E);
}

Java

public static double code(double x) {
	return (-1.0 + ((x * x) + 2.0)) / Math.E;
}

Python

def code(x):
	return (-1.0 + ((x * x) + 2.0)) / math.e

Julia

function code(x)
	return Float64(Float64(-1.0 + Float64(Float64(x * x) + 2.0)) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 + ((x * x) + 2.0)) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[(-1.0 + N[(N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $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. Taylor expanded in x around 0 74.5%

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

   1. distribute-rgt1-in 74.5%

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

   2. unpow2 74.5%

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

   3. fma-undefine 74.5%

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

   4. metadata-eval 74.5%

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

   5. rec-exp 74.5%

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

   6. e-exp-1 74.5%

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

   7. associate-*r/ 74.5%

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

   8. metadata-eval 74.5%

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

   9. distribute-rgt-neg-in 74.5%

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

   10. *-commutative 74.5%

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

   11. neg-mul-1 74.5%

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

   12. remove-double-neg 74.5%

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

7. Simplified 74.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{e}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 74.5%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right)\right)\right)}}{e} \]

   2. expm1-undefine 74.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right)\right)} - 1}}{e} \]

9. Applied egg-rr 74.5%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right)\right)} - 1}}{e} \]
10. Step-by-step derivation

    1. sub-neg 74.5%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right)\right)} + \left(-1\right)}}{e} \]

    2. metadata-eval 74.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right)\right)} + \color{blue}{-1}}{e} \]

    3. +-commutative 74.5%

       \[\leadsto \frac{\color{blue}{-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, 1\right)\right)}}}{e} \]

    4. log1p-undefine 74.5%

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

    5. rem-exp-log 74.5%

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

    6. fma-undefine 74.5%

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

    7. unpow2 74.5%

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

    8. +-commutative 74.5%

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

    9. associate-+r+ 74.5%

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

    10. metadata-eval 74.5%

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

11. Simplified 74.5%

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

    1. unpow2 74.5%

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

13. Applied egg-rr 74.5%

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

14. Final simplification 74.5%

    \[\leadsto \frac{-1 + \left(x \cdot x + 2\right)}{e} \]
15. Add Preprocessing

Alternative 5: 51.2% 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. Taylor expanded in x around 0 74.5%

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

   1. distribute-rgt1-in 74.5%

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

   2. unpow2 74.5%

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

   3. fma-undefine 74.5%

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

   4. metadata-eval 74.5%

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

   5. rec-exp 74.5%

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

   6. e-exp-1 74.5%

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

   7. associate-*r/ 74.5%

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

   8. metadata-eval 74.5%

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

   9. distribute-rgt-neg-in 74.5%

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

   10. *-commutative 74.5%

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

   11. neg-mul-1 74.5%

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

   12. remove-double-neg 74.5%

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

7. Simplified 74.5%

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

8. Taylor expanded in x around 0 47.2%

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

Reproduce

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