exp neg sub

Average Accuracy: 100.0% → 100.0%

Time: 3.2s

Precision: binary64

Cost: 6976

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   Step-by-step derivation

   [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%

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

   [Start] 100.0	\[ e^{x \cdot x + -1} \]
   difference-of-sqr--1 [=>] 100.0	\[ e^{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   sub-neg [=>] 100.0	\[ e^{\left(x + 1\right) \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 100.0	\[ e^{\left(x + 1\right) \cdot \left(x + \color{blue}{-1}\right)} \]

4. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ e^{\left(x + 1\right) \cdot \left(x + -1\right)} \]
   *-commutative [=>] 100.0	\[ e^{\color{blue}{\left(x + -1\right) \cdot \left(x + 1\right)}} \]
   +-commutative [=>] 100.0	\[ e^{\left(x + -1\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   distribute-lft-in [=>] 100.0	\[ e^{\color{blue}{\left(x + -1\right) \cdot 1 + \left(x + -1\right) \cdot x}} \]
   *-commutative [<=] 100.0	\[ e^{\color{blue}{1 \cdot \left(x + -1\right)} + \left(x + -1\right) \cdot x} \]
   *-un-lft-identity [<=] 100.0	\[ e^{\color{blue}{\left(x + -1\right)} + \left(x + -1\right) \cdot x} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.95:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	6720

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

Alternative 3
Accuracy	50.8%
Cost	6464

\[e^{-1} \]

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))