exp-w (used to crash)

Average Error: 0.3 → 0.3

Time: 17.2s

Precision: binary64

Cost: 64832

Math

\[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

↓

\[\begin{array}{l} t_0 := {\left(e^{w}\right)}^{0.25}\\ e^{-w} \cdot {\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{t_0}\right)}^{\left(2 \cdot \frac{t_0}{2}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

↓

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow (exp w) 0.25)))
   (* (exp (- w)) (pow (pow (pow l (sqrt (exp w))) t_0) (* 2.0 (/ t_0 2.0))))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

↓

double code(double w, double l) {
	double t_0 = pow(exp(w), 0.25);
	return exp(-w) * pow(pow(pow(l, sqrt(exp(w))), t_0), (2.0 * (t_0 / 2.0)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

↓

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = exp(w) ** 0.25d0
    code = exp(-w) * (((l ** sqrt(exp(w))) ** t_0) ** (2.0d0 * (t_0 / 2.0d0)))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

↓

public static double code(double w, double l) {
	double t_0 = Math.pow(Math.exp(w), 0.25);
	return Math.exp(-w) * Math.pow(Math.pow(Math.pow(l, Math.sqrt(Math.exp(w))), t_0), (2.0 * (t_0 / 2.0)));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

↓

def code(w, l):
	t_0 = math.pow(math.exp(w), 0.25)
	return math.exp(-w) * math.pow(math.pow(math.pow(l, math.sqrt(math.exp(w))), t_0), (2.0 * (t_0 / 2.0)))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

↓

function code(w, l)
	t_0 = exp(w) ^ 0.25
	return Float64(exp(Float64(-w)) * (((l ^ sqrt(exp(w))) ^ t_0) ^ Float64(2.0 * Float64(t_0 / 2.0))))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

↓

function tmp = code(w, l)
	t_0 = exp(w) ^ 0.25;
	tmp = exp(-w) * (((l ^ sqrt(exp(w))) ^ t_0) ^ (2.0 * (t_0 / 2.0)));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w_, l_] := Block[{t$95$0 = N[Power[N[Exp[w], $MachinePrecision], 0.25], $MachinePrecision]}, N[(N[Exp[(-w)], $MachinePrecision] * N[Power[N[Power[N[Power[l, N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision], N[(2.0 * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.3

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Taylor expanded in l around 0 4.6

   \[\leadsto e^{-w} \cdot \color{blue}{e^{\log \ell \cdot e^{w}}} \]

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.3

   \[\leadsto e^{-w} \cdot {\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left(e^{w \cdot 0.5}\right)}} \]

5. Applied egg-rr 0.7

   \[\leadsto e^{-w} \cdot \color{blue}{\left({\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(\frac{{\left(e^{w}\right)}^{0.25}}{2}\right)} \cdot {\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(\frac{{\left(e^{w}\right)}^{0.25}}{2}\right)}\right)} \]

6. Simplified 0.3

   \[\leadsto e^{-w} \cdot \color{blue}{{\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(2 \cdot \frac{{\left(e^{w}\right)}^{0.25}}{2}\right)}} \]
   Proof

   [Start] 0.7	\[ e^{-w} \cdot \left({\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(\frac{{\left(e^{w}\right)}^{0.25}}{2}\right)} \cdot {\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(\frac{{\left(e^{w}\right)}^{0.25}}{2}\right)}\right) \]
   pow-sqr [=>] 0.3	\[ e^{-w} \cdot \color{blue}{{\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(2 \cdot \frac{{\left(e^{w}\right)}^{0.25}}{2}\right)}} \]

7. Final simplification 0.3

   \[\leadsto e^{-w} \cdot {\left({\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.25}\right)}\right)}^{\left(2 \cdot \frac{{\left(e^{w}\right)}^{0.25}}{2}\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	38912

\[e^{-w} \cdot {\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left(e^{w \cdot 0.5}\right)} \]

Alternative 2
Error	0.3
Cost	32640

\[\begin{array}{l} t_0 := e^{w \cdot 0.5}\\ e^{-w} \cdot {\left({\ell}^{t_0}\right)}^{t_0} \end{array} \]

Alternative 3
Error	0.3
Cost	19456

\[\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]

Alternative 4
Error	1.2
Cost	13376

\[\frac{\ell \cdot \left(1 + w \cdot \log \ell\right)}{e^{w}} \]

Alternative 5
Error	1.7
Cost	6784

\[\ell \cdot \frac{-1}{-e^{w}} \]

Alternative 6
Error	1.7
Cost	6660

\[\begin{array}{l} \mathbf{if}\;w \leq 360:\\ \;\;\;\;\ell \cdot \left(1 - w\right) + \left(w \cdot w\right) \cdot \left(\ell \cdot 0.5 + w \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	6592

\[\frac{\ell}{e^{w}} \]

Alternative 8
Error	8.2
Cost	1348

\[\begin{array}{l} t_0 := \ell \cdot \left(1 - w\right)\\ \mathbf{if}\;w \leq 0.08:\\ \;\;\;\;t_0 + \left(w \cdot w\right) \cdot \left(\ell \cdot 0.5 + w \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \end{array} \]

Alternative 9
Error	8.2
Cost	708

\[\begin{array}{l} t_0 := \ell \cdot \left(1 - w\right)\\ \mathbf{if}\;w \leq 0.105:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \end{array} \]

Alternative 10
Error	13.6
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023045 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))