exp neg sub

Average Error: 0.0 → 0.0

Time: 3.0s

Precision: binary64

Cost: 38784

Math

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

↓

\[{\left(e^{2}\right)}^{\left(x \cdot \log \left(\sqrt{e^{x}}\right)\right)} \cdot e^{-1} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (pow (exp 2.0) (* x (log (sqrt (exp x))))) (exp -1.0)))

C

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

↓

double code(double x) {
	return pow(exp(2.0), (x * log(sqrt(exp(x))))) * exp(-1.0);
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(2.0d0) ** (x * log(sqrt(exp(x))))) * exp((-1.0d0))
end function

Java

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

↓

public static double code(double x) {
	return Math.pow(Math.exp(2.0), (x * Math.log(Math.sqrt(Math.exp(x))))) * Math.exp(-1.0);
}

Python

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

↓

def code(x):
	return math.pow(math.exp(2.0), (x * math.log(math.sqrt(math.exp(x))))) * math.exp(-1.0)

Julia

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

↓

function code(x)
	return Float64((exp(2.0) ^ Float64(x * log(sqrt(exp(x))))) * exp(-1.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (exp(2.0) ^ (x * log(sqrt(exp(x))))) * exp(-1.0);
end

Wolfram

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

↓

code[x_] := N[(N[Power[N[Exp[2.0], $MachinePrecision], N[(x * N[Log[N[Sqrt[N[Exp[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[-1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ e^{-\left(1 - x \cdot x\right)} \]
   neg-sub0 [=>] 0.0	\[ e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]
   associate--r- [=>] 0.0	\[ e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]
   metadata-eval [=>] 0.0	\[ e^{\color{blue}{-1} + x \cdot x} \]
   +-commutative [=>] 0.0	\[ e^{\color{blue}{x \cdot x + -1}} \]

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

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

5. Simplified 0.0

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

   [Start] 0.0	\[ \left({\left(\sqrt{e^{x}}\right)}^{x} \cdot {\left(\sqrt{e^{x}}\right)}^{x}\right) \cdot e^{-1} \]
   pow-sqr [=>] 0.0	\[ \color{blue}{{\left(\sqrt{e^{x}}\right)}^{\left(2 \cdot x\right)}} \cdot e^{-1} \]
   count-2 [<=] 0.0	\[ {\left(\sqrt{e^{x}}\right)}^{\color{blue}{\left(x + x\right)}} \cdot e^{-1} \]

6. Taylor expanded in x around inf 0.0

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

7. Simplified 0.0

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

   [Start] 0.0	\[ e^{2 \cdot \left(\log \left(\sqrt{e^{x}}\right) \cdot x\right)} \cdot e^{-1} \]
   exp-prod [=>] 0.0	\[ \color{blue}{{\left(e^{2}\right)}^{\left(\log \left(\sqrt{e^{x}}\right) \cdot x\right)}} \cdot e^{-1} \]
   *-commutative [=>] 0.0	\[ {\left(e^{2}\right)}^{\color{blue}{\left(x \cdot \log \left(\sqrt{e^{x}}\right)\right)}} \cdot e^{-1} \]

8. Final simplification 0.0

   \[\leadsto {\left(e^{2}\right)}^{\left(x \cdot \log \left(\sqrt{e^{x}}\right)\right)} \cdot e^{-1} \]

Alternatives

Alternative 1
Error	0.0
Cost	19712

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

Alternative 2
Error	0.0
Cost	6720

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

Alternative 3
Error	0.9
Cost	6464

\[e^{-1} \]

Reproduce

herbie shell --seed 2023017 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))