exp-w (used to crash)

Average Error: 0.3 → 0.3

Time: 15.7s

Precision: binary64

Cost: 45248

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{{\ell}^{\left(e^{w}\right)}}{t_0 \cdot {t_0}^{2}} \end{array} \]

FPCore

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

↓

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (exp w)))) (/ (pow l (exp w)) (* t_0 (pow 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 = cbrt(exp(w));
	return pow(l, exp(w)) / (t_0 * pow(t_0, 2.0));
}

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.cbrt(Math.exp(w));
	return Math.pow(l, Math.exp(w)) / (t_0 * Math.pow(t_0, 2.0));
}

Julia

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

↓

function code(w, l)
	t_0 = cbrt(exp(w))
	return Float64((l ^ exp(w)) / Float64(t_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], 1/3], $MachinePrecision]}, N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   Proof

   [Start] 0.3	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   exp-neg [=>] 0.3	\[ \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]
   associate-*l/ [=>] 0.3	\[ \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   *-lft-identity [=>] 0.3	\[ \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

4. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2} \cdot \sqrt[3]{e^{w}}}} \]
   Proof

   [Start] 0.3	\[ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   associate-*l/ [=>] 0.3	\[ \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
   *-lft-identity [=>] 0.3	\[ \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
   associate-/l/ [=>] 0.3	\[ \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2} \cdot \sqrt[3]{e^{w}}}} \]

5. Final simplification 0.3

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot {\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

Alternatives

Alternative 1
Error	0.3
Cost	19456

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

Alternative 2
Error	1.4
Cost	13376

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

Alternative 3
Error	1.6
Cost	13376

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

Alternative 4
Error	1.8
Cost	6656

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

Alternative 5
Error	1.8
Cost	6592

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

Alternative 6
Error	2.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \]

Alternative 7
Error	13.3
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023059 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))