exp neg sub

Average Accuracy: 100.0% → 99.9%

Time: 4.5s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

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

   [Start] 100.0	\[ e^{x \cdot x + -1} \]
   difference-of-sqr--1 [=>] 99.9	\[ e^{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   sub-neg [=>] 99.9	\[ e^{\left(x + 1\right) \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 99.9	\[ e^{\left(x + 1\right) \cdot \left(x + \color{blue}{-1}\right)} \]

4. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ e^{\left(x + 1\right) \cdot \left(x + -1\right)} \]
   *-commutative [=>] 99.9	\[ e^{\color{blue}{\left(x + -1\right) \cdot \left(x + 1\right)}} \]
   +-commutative [=>] 99.9	\[ e^{\left(x + -1\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   distribute-lft-in [=>] 99.9	\[ e^{\color{blue}{\left(x + -1\right) \cdot 1 + \left(x + -1\right) \cdot x}} \]
   *-commutative [<=] 99.9	\[ e^{\color{blue}{1 \cdot \left(x + -1\right)} + \left(x + -1\right) \cdot x} \]
   *-un-lft-identity [<=] 99.9	\[ e^{\color{blue}{\left(x + -1\right)} + \left(x + -1\right) \cdot x} \]
   metadata-eval [<=] 99.9	\[ e^{\left(x + \color{blue}{\left(-1\right)}\right) + \left(x + -1\right) \cdot x} \]
   sub-neg [<=] 99.9	\[ e^{\color{blue}{\left(x - 1\right)} + \left(x + -1\right) \cdot x} \]
   associate-+l- [=>] 99.9	\[ e^{\color{blue}{x - \left(1 - \left(x + -1\right) \cdot x\right)}} \]
   *-commutative [=>] 99.9	\[ e^{x - \left(1 - \color{blue}{x \cdot \left(x + -1\right)}\right)} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

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

Alternative 2
Accuracy	98.4%
Cost	6464

\[e^{-1} \]

Alternative 3
Accuracy	17.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))