Average Error: 0.0 → 0.0
Time: 2.9s
Precision: binary64
Cost: 19712
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[e^{-1} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)} \]
(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))
↓
(FPCore (x) :precision binary64 (* (exp -1.0) (pow (exp (+ x x)) (/ x 2.0))))
double code(double x) {
	return exp(-(1.0 - (x * x)));
}
↓
double code(double x) {
	return exp(-1.0) * pow(exp((x + x)), (x / 2.0));
}

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((-1.0d0)) * (exp((x + x)) ** (x / 2.0d0))
end function

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}
↓
public static double code(double x) {
	return Math.exp(-1.0) * Math.pow(Math.exp((x + x)), (x / 2.0));
}

def code(x):
	return math.exp(-(1.0 - (x * x)))
↓
def code(x):
	return math.exp(-1.0) * math.pow(math.exp((x + x)), (x / 2.0))

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end
↓
function code(x)
	return Float64(exp(-1.0) * (exp(Float64(x + x)) ^ Float64(x / 2.0)))
end

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end
↓
function tmp = code(x)
	tmp = exp(-1.0) * (exp((x + x)) ^ (x / 2.0));
end

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]
↓
code[x_] := N[(N[Exp[-1.0], $MachinePrecision] * N[Power[N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

e^{-\left(1 - x \cdot x\right)}
↓
e^{-1} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\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

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

  4. Applied egg-rr0.0

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

  5. Applied egg-rr0.0

    \[\leadsto e^{-1} \cdot \color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(\frac{x}{2}\right)}} \]

  6. Simplified0.0

    \[\leadsto e^{-1} \cdot \color{blue}{{\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}} \]
    Proof
    (pow.f64 (exp.f64 (+.f64 x x)) (/.f64 x 2)): 0 points increase in error, 0 points decrease in error
    (pow.f64 (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 x) (exp.f64 x))) (/.f64 x 2)): 0 points increase in error, 1 points decrease in error
    (pow.f64 (Rewrite<= unpow2_binary64 (pow.f64 (exp.f64 x) 2)) (/.f64 x 2)): 0 points increase in error, 0 points decrease in error

  7. Final simplification0.0

    \[\leadsto e^{-1} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)} \]

Alternatives

Alternative 1
Error0.0
Cost19456
\[e^{-1} \cdot {\left(e^{x}\right)}^{x} \]
Alternative 2
Error0.0
Cost13184
\[{\left(e^{x + 1}\right)}^{\left(-1 + x\right)} \]
Alternative 3
Error0.0
Cost6720
\[e^{-1 + x \cdot x} \]
Alternative 4
Error0.9
Cost6464
\[e^{-1} \]

Error

Reproduce

herbie shell --seed 2022318 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))