Average Error: 0.0 → 0.0
Time: 3.9s
Precision: binary64
Cost: 26112
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[{\left(e^{x}\right)}^{\left(x + -1\right)} \cdot {e}^{\left(x + -1\right)} \]
(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))
↓
(FPCore (x)
 :precision binary64
 (* (pow (exp x) (+ x -1.0)) (pow E (+ x -1.0))))
double code(double x) {
	return exp(-(1.0 - (x * x)));
}
↓
double code(double x) {
	return pow(exp(x), (x + -1.0)) * pow(((double) M_E), (x + -1.0));
}

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), (x + -1.0)) * Math.pow(Math.E, (x + -1.0));
}

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

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end
↓
function code(x)
	return Float64((exp(x) ^ Float64(x + -1.0)) * (exp(1) ^ Float64(x + -1.0)))
end

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end
↓
function tmp = code(x)
	tmp = (exp(x) ^ (x + -1.0)) * (2.71828182845904523536 ^ (x + -1.0));
end

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

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
    Proof
    (exp.f64 (+.f64 (*.f64 x x) -1)): 0 points increase in error, 0 points decrease in error
    (exp.f64 (Rewrite<= +-commutative_binary64 (+.f64 -1 (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
    (exp.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
    (exp.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
    (exp.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.1

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

  5. Simplified0.0

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{\left(-1 + x\right)} \cdot {e}^{\left(-1 + x\right)}} \]
    Proof
    (*.f64 (pow.f64 (exp.f64 x) (+.f64 -1 x)) (pow.f64 (E.f64) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 (exp.f64 x) (Rewrite<= +-commutative_binary64 (+.f64 x -1))) (pow.f64 (E.f64) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 (exp.f64 x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 x -1)))) (pow.f64 (E.f64) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 (exp.f64 x) (*.f64 (Rewrite<= metadata-eval (*.f64 2 1/2)) (+.f64 x -1))) (pow.f64 (E.f64) (+.f64 -1 x))): 6 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 (exp.f64 x) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 1/2 (+.f64 x -1))))) (pow.f64 (E.f64) (+.f64 -1 x))): 0 points increase in error, 6 points decrease in error
    (*.f64 (pow.f64 (exp.f64 x) (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x -1) 1/2)))) (pow.f64 (E.f64) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)))) (pow.f64 (E.f64) (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2))) (pow.f64 (E.f64) (Rewrite<= +-commutative_binary64 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2))) (pow.f64 (E.f64) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 x -1))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2))) (pow.f64 (E.f64) (*.f64 (Rewrite<= metadata-eval (*.f64 2 1/2)) (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2))) (pow.f64 (E.f64) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 1/2 (+.f64 x -1)))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2))) (pow.f64 (E.f64) (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x -1) 1/2))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (exp.f64 x) (*.f64 (+.f64 x -1) 1/2))) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (E.f64) (*.f64 (+.f64 x -1) 1/2)) (pow.f64 (E.f64) (*.f64 (+.f64 x -1) 1/2))))): 0 points increase in error, 0 points decrease in error

  6. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost13184
\[{\left(e^{x + -1}\right)}^{\left(x + 1\right)} \]
Alternative 2
Error0.0
Cost6720
\[e^{-1 + x \cdot x} \]
Alternative 3
Error1.0
Cost6464
\[e^{-1} \]
Alternative 4
Error52.6
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))