exp neg sub

Average Error: 0.0 → 0.0

Time: 2.9s

Precision: binary64

Cost: 19712

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

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

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13184

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

Alternative 2
Error	0.0
Cost	6720

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

Alternative 3
Error	1.0
Cost	6464

\[e^{-1} \]

Alternative 4
Error	52.6
Cost	320

\[1 + x \cdot x \]

Alternative 5
Error	52.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))