exp neg sub

Average Error: 100.0% → 99.9%

Time: 8.2s

Precision: binary64

Cost: 19776.00

Math

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

↓

\[{\left({\left(e^{0.5 + 0.5 \cdot x}\right)}^{\left(x + -1\right)}\right)}^{2} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (pow (pow (exp (+ 0.5 (* 0.5 x))) (+ x -1.0)) 2.0))

C

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

↓

double code(double x) {
	return pow(pow(exp((0.5 + (0.5 * x))), (x + -1.0)), 2.0);
}

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((0.5d0 + (0.5d0 * x))) ** (x + (-1.0d0))) ** 2.0d0
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.pow(Math.exp((0.5 + (0.5 * x))), (x + -1.0)), 2.0);
}

Python

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

↓

def code(x):
	return math.pow(math.pow(math.exp((0.5 + (0.5 * x))), (x + -1.0)), 2.0)

Julia

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

↓

function code(x)
	return (exp(Float64(0.5 + Float64(0.5 * x))) ^ Float64(x + -1.0)) ^ 2.0
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (exp((0.5 + (0.5 * x))) ^ (x + -1.0)) ^ 2.0;
end

Wolfram

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

↓

code[x_] := N[Power[N[Power[N[Exp[N[(0.5 + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x + -1.0), $MachinePrecision]], $MachinePrecision], 2.0], $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

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

4. Applied egg-rr 99.9

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

5. Taylor expanded in x around inf 99.9

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

6. Applied egg-rr 99.9

   \[\leadsto {\left({\color{blue}{\left(e^{0.5 + x \cdot 0.5}\right)}}^{\left(x - 1\right)}\right)}^{2} \]

7. Final simplification 99.9

   \[\leadsto {\left({\left(e^{0.5 + 0.5 \cdot x}\right)}^{\left(x + -1\right)}\right)}^{2} \]

Alternatives

Alternative 1
Error	100.0%
Cost	6720.00

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

Alternative 2
Error	98.7%
Cost	6464.00

\[e^{-1} \]

Reproduce

herbie shell --seed 2023093 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))