exp neg sub

Average Accuracy: 100.0% → 99.9%

Time: 3.4s

Precision: binary64

Cost: 92032

Math

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

↓

\[\begin{array}{l} t_0 := {\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)}\\ t_1 := {\left(\sqrt{e^{x}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)}\\ \left(\left(t_1 \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{4}\right)}\right) \cdot \left(t_1 \cdot \left(t_0 \cdot t_0\right)\right)\right) \cdot {e}^{\left(x + -1\right)} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) (/ (+ x -1.0) 8.0)))
        (t_1 (pow (sqrt (exp x)) (* (+ x -1.0) 0.5))))
   (*
    (* (* t_1 (pow (exp x) (/ (+ x -1.0) 4.0))) (* t_1 (* t_0 t_0)))
    (pow E (+ x -1.0)))))

C

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

↓

double code(double x) {
	double t_0 = pow(exp(x), ((x + -1.0) / 8.0));
	double t_1 = pow(sqrt(exp(x)), ((x + -1.0) * 0.5));
	return ((t_1 * pow(exp(x), ((x + -1.0) / 4.0))) * (t_1 * (t_0 * t_0))) * pow(((double) M_E), (x + -1.0));
}

Java

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

↓

public static double code(double x) {
	double t_0 = Math.pow(Math.exp(x), ((x + -1.0) / 8.0));
	double t_1 = Math.pow(Math.sqrt(Math.exp(x)), ((x + -1.0) * 0.5));
	return ((t_1 * Math.pow(Math.exp(x), ((x + -1.0) / 4.0))) * (t_1 * (t_0 * t_0))) * Math.pow(Math.E, (x + -1.0));
}

Python

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

↓

def code(x):
	t_0 = math.pow(math.exp(x), ((x + -1.0) / 8.0))
	t_1 = math.pow(math.sqrt(math.exp(x)), ((x + -1.0) * 0.5))
	return ((t_1 * math.pow(math.exp(x), ((x + -1.0) / 4.0))) * (t_1 * (t_0 * t_0))) * math.pow(math.e, (x + -1.0))

Julia

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

↓

function code(x)
	t_0 = exp(x) ^ Float64(Float64(x + -1.0) / 8.0)
	t_1 = sqrt(exp(x)) ^ Float64(Float64(x + -1.0) * 0.5)
	return Float64(Float64(Float64(t_1 * (exp(x) ^ Float64(Float64(x + -1.0) / 4.0))) * Float64(t_1 * Float64(t_0 * t_0))) * (exp(1) ^ Float64(x + -1.0)))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = exp(x) ^ ((x + -1.0) / 8.0);
	t_1 = sqrt(exp(x)) ^ ((x + -1.0) * 0.5);
	tmp = ((t_1 * (exp(x) ^ ((x + -1.0) / 4.0))) * (t_1 * (t_0 * t_0))) * (2.71828182845904523536 ^ (x + -1.0));
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] / 8.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[N[Exp[x], $MachinePrecision]], $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$1 * N[Power[N[Exp[x], $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[E, N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ e^{-\left(1 - x \cdot x\right)} \]
   neg-sub0 [=>] 100.0	\[ e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]
   associate--r- [=>] 100.0	\[ e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]
   metadata-eval [=>] 100.0	\[ e^{\color{blue}{-1} + x \cdot x} \]
   +-commutative [=>] 100.0	\[ e^{\color{blue}{x \cdot x + -1}} \]

3. Applied egg-rr 99.9%

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

   [Start] 100.0	\[ e^{x \cdot x + -1} \]
   difference-of-sqr--1 [=>] 99.9	\[ e^{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   exp-prod [=>] 99.9	\[ \color{blue}{{\left(e^{x + 1}\right)}^{\left(x - 1\right)}} \]
   sub-neg [=>] 99.9	\[ {\left(e^{x + 1}\right)}^{\color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 99.9	\[ {\left(e^{x + 1}\right)}^{\left(x + \color{blue}{-1}\right)} \]

4. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ {\left(e^{x + 1}\right)}^{\left(x + -1\right)} \]
   sqr-pow [=>] 100.0	\[ \color{blue}{{\left(e^{x + 1}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{x + 1}\right)}^{\left(\frac{x + -1}{2}\right)}} \]
   exp-sum [=>] 100.0	\[ {\color{blue}{\left(e^{x} \cdot e^{1}\right)}}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{x + 1}\right)}^{\left(\frac{x + -1}{2}\right)} \]
   unpow-prod-down [=>] 99.9	\[ \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right)} \cdot {\left(e^{x + 1}\right)}^{\left(\frac{x + -1}{2}\right)} \]
   exp-sum [=>] 99.9	\[ \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \cdot {\color{blue}{\left(e^{x} \cdot e^{1}\right)}}^{\left(\frac{x + -1}{2}\right)} \]
   unpow-prod-down [=>] 99.9	\[ \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \cdot \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right)} \]
   swap-sqr [=>] 99.9	\[ \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \cdot \left({\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right)} \]
   div-inv [=>] 99.9	\[ \left({\left(e^{x}\right)}^{\color{blue}{\left(\left(x + -1\right) \cdot \frac{1}{2}\right)}} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \cdot \left({\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \left({\left(e^{x}\right)}^{\left(\left(x + -1\right) \cdot \color{blue}{0.5}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \cdot \left({\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \]
   div-inv [=>] 99.9	\[ \left({\left(e^{x}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\left(x + -1\right) \cdot \frac{1}{2}\right)}}\right) \cdot \left({\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \left({\left(e^{x}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\left(x + -1\right) \cdot \color{blue}{0.5}\right)}\right) \cdot \left({\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{x + -1}{2}\right)}\right) \]

5. Simplified 99.9%

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

   [Start] 99.9	\[ \left({\left(e^{x}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   pow-sqr [=>] 99.9	\[ \color{blue}{{\left(e^{x}\right)}^{\left(2 \cdot \left(\left(x + -1\right) \cdot 0.5\right)\right)}} \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   *-commutative [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(2 \cdot \color{blue}{\left(0.5 \cdot \left(x + -1\right)\right)}\right)} \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   associate-*r* [=>] 99.9	\[ {\left(e^{x}\right)}^{\color{blue}{\left(\left(2 \cdot 0.5\right) \cdot \left(x + -1\right)\right)}} \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   metadata-eval [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(\color{blue}{1} \cdot \left(x + -1\right)\right)} \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   *-lft-identity [=>] 99.9	\[ {\left(e^{x}\right)}^{\color{blue}{\left(x + -1\right)}} \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   +-commutative [=>] 99.9	\[ {\left(e^{x}\right)}^{\color{blue}{\left(-1 + x\right)}} \cdot \left({e}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {e}^{\left(\left(x + -1\right) \cdot 0.5\right)}\right) \]
   pow-sqr [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot \color{blue}{{e}^{\left(2 \cdot \left(\left(x + -1\right) \cdot 0.5\right)\right)}} \]
   *-commutative [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\left(2 \cdot \color{blue}{\left(0.5 \cdot \left(x + -1\right)\right)}\right)} \]
   associate-*r* [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\color{blue}{\left(\left(2 \cdot 0.5\right) \cdot \left(x + -1\right)\right)}} \]
   metadata-eval [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\left(\color{blue}{1} \cdot \left(x + -1\right)\right)} \]
   *-lft-identity [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\color{blue}{\left(x + -1\right)}} \]
   +-commutative [=>] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\color{blue}{\left(-1 + x\right)}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)} \cdot {e}^{\left(-1 + x\right)} \]
   Proof

   [Start] 99.9	\[ {\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\left(-1 + x\right)} \]
   sqr-pow [=>] 99.9	\[ \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{2}\right)}\right)} \cdot {e}^{\left(-1 + x\right)} \]
   add-sqr-sqrt [=>] 99.9	\[ \left({\color{blue}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{2}\right)}\right) \cdot {e}^{\left(-1 + x\right)} \]
   unpow-prod-down [=>] 99.9	\[ \left(\color{blue}{\left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{2}\right)}\right) \cdot {e}^{\left(-1 + x\right)} \]
   sqr-pow [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)}\right) \cdot \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right)}\right) \cdot {e}^{\left(-1 + x\right)} \]
   unswap-sqr [=>] 99.9	\[ \color{blue}{\left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right)\right)} \cdot {e}^{\left(-1 + x\right)} \]
   div-inv [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\color{blue}{\left(\left(-1 + x\right) \cdot \frac{1}{2}\right)}} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot \color{blue}{0.5}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   associate-/l/ [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\frac{-1 + x}{2 \cdot 2}\right)}}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{\color{blue}{4}}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{2}}{2}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]

7. Applied egg-rr 99.9%

   \[\leadsto \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   Proof

   [Start] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   add-sqr-sqrt [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\color{blue}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}^{\left(\frac{-1 + x}{4}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   pow-prod-down [<=] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \color{blue}{\left({\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)}\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   sqrt-pow2 [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left(\color{blue}{{\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{4}}{2}\right)}} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [<=] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{4}}{\color{blue}{\sqrt{4}}}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   associate-/l/ [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\color{blue}{\left(\frac{-1 + x}{\sqrt{4} \cdot 4}\right)}} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   +-commutative [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{\color{blue}{x + -1}}{\sqrt{4} \cdot 4}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{\color{blue}{2} \cdot 4}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{\color{blue}{8}}\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\frac{-1 + x}{4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   sqrt-pow2 [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot \color{blue}{{\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{4}}{2}\right)}}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [<=] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{-1 + x}{4}}{\color{blue}{\sqrt{4}}}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   associate-/l/ [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\frac{-1 + x}{\sqrt{4} \cdot 4}\right)}}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   +-commutative [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\color{blue}{x + -1}}{\sqrt{4} \cdot 4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{\color{blue}{2} \cdot 4}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]
   metadata-eval [=>] 99.9	\[ \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{-1 + x}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{\color{blue}{8}}\right)}\right)\right)\right) \cdot {e}^{\left(-1 + x\right)} \]

8. Final simplification 99.9%

   \[\leadsto \left(\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{4}\right)}\right) \cdot \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(x + -1\right) \cdot 0.5\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x + -1}{8}\right)}\right)\right)\right) \cdot {e}^{\left(x + -1\right)} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13184

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

Alternative 2
Accuracy	100.0%
Cost	6720

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

Alternative 3
Accuracy	98.3%
Cost	6464

\[e^{-1} \]

Reproduce

herbie shell --seed 2023136 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))