exp neg sub

Average Accuracy: 100.0% → 100.0%

Time: 3.4s

Precision: binary64

Cost: 19712

Math

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

↓

\[\begin{array}{l} t_0 := e^{x + 1}\\ \frac{{t_0}^{x}}{t_0} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (+ x 1.0)))) (/ (pow t_0 x) t_0)))

C

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

↓

double code(double x) {
	double t_0 = exp((x + 1.0));
	return pow(t_0, x) / t_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
    real(8) :: t_0
    t_0 = exp((x + 1.0d0))
    code = (t_0 ** x) / t_0
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

↓

public static double code(double x) {
	double t_0 = Math.exp((x + 1.0));
	return Math.pow(t_0, x) / t_0;
}

Python

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

↓

def code(x):
	t_0 = math.exp((x + 1.0))
	return math.pow(t_0, x) / t_0

Julia

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

↓

function code(x)
	t_0 = exp(Float64(x + 1.0))
	return Float64((t_0 ^ x) / t_0)
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = exp((x + 1.0));
	tmp = (t_0 ^ x) / t_0;
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[Exp[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[t$95$0, x], $MachinePrecision] / t$95$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 100.0%

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

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13184

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

Alternative 2
Accuracy	100.0%
Cost	6720

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

Alternative 3
Accuracy	98.7%
Cost	6464

\[e^{-1} \]

Alternative 4
Accuracy	17.8%
Cost	320

\[1 + x \cdot x \]

Alternative 5
Accuracy	17.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))