exp neg sub

Average Accuracy: 100.0% → 100.0%

Time: 7.3s

Precision: binary64

Cost: 19456

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(x):
	return math.pow(math.exp(x), 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(x) ^ x) * exp(-1.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (exp(x) ^ 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[x], $MachinePrecision], x], $MachinePrecision] * N[Exp[-1.0], $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 100.0%

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

   [Start] 100.0	\[ e^{x \cdot x + -1} \]
   exp-sum [=>] 100.0	\[ \color{blue}{e^{x \cdot x} \cdot e^{-1}} \]
   exp-prod [=>] 100.0	\[ \color{blue}{{\left(e^{x}\right)}^{x}} \cdot e^{-1} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	12992

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

Alternative 2
Accuracy	100.0%
Cost	6720

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

Alternative 3
Accuracy	98.6%
Cost	6464

\[e^{-1} \]

Alternative 4
Accuracy	17.8%
Cost	320

\[x \cdot x + 1 \]

Alternative 5
Accuracy	17.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023146 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))