exp neg sub

Average Accuracy: 100.0% → 99.9%

Time: 3.5s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return exp((x + -1.0)) * exp((x * (x + -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((x + (-1.0d0))) * exp((x * (x + (-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.exp((x + -1.0)) * Math.exp((x * (x + -1.0)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

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

6. Taylor expanded in x around -inf 99.9%

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

7. Taylor expanded in x around inf 99.9%

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

8. Simplified 99.9%

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

   [Start] 99.9	\[ e^{-1 \cdot \left(\log e \cdot \left(1 - x\right)\right)} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   mul-1-neg [=>] 99.9	\[ e^{\color{blue}{-\log e \cdot \left(1 - x\right)}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   log-E [=>] 99.9	\[ e^{-\color{blue}{1} \cdot \left(1 - x\right)} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   *-lft-identity [=>] 99.9	\[ e^{-\color{blue}{\left(1 - x\right)}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   exp-neg [=>] 99.9	\[ \color{blue}{\frac{1}{e^{1 - x}}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   div-exp [<=] 99.9	\[ \frac{1}{\color{blue}{\frac{e^{1}}{e^{x}}}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   metadata-eval [<=] 99.9	\[ \frac{1}{\frac{e^{\color{blue}{--1}}}{e^{x}}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   rec-exp [<=] 99.9	\[ \frac{1}{\frac{\color{blue}{\frac{1}{e^{-1}}}}{e^{x}}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   associate-/r* [<=] 99.9	\[ \frac{1}{\color{blue}{\frac{1}{e^{-1} \cdot e^{x}}}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   exp-sum [<=] 99.9	\[ \frac{1}{\frac{1}{\color{blue}{e^{-1 + x}}}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   associate-/l* [<=] 99.9	\[ \color{blue}{\frac{1 \cdot e^{-1 + x}}{1}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   *-lft-identity [=>] 99.9	\[ \frac{\color{blue}{e^{-1 + x}}}{1} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   /-rgt-identity [=>] 99.9	\[ \color{blue}{e^{-1 + x}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]
   +-commutative [=>] 99.9	\[ e^{\color{blue}{x + -1}} \cdot e^{-1 \cdot \left(\left(1 + -1 \cdot x\right) \cdot x\right)} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

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

Alternative 2
Accuracy	98.5%
Cost	6464

\[e^{-1} \]

Reproduce

herbie shell --seed 2023143 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))