exp-w (used to crash)

Average Error: 0.3 → 0.3

Time: 16.1s

Precision: binary64

Cost: 58112

Math

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

↓

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

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 (pow (cbrt t_0) 3.0) 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(pow(cbrt(t_0), 3.0), 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(Math.pow(Math.cbrt(t_0), 3.0), 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(Float64((l ^ exp(w)) / t_0) / ((cbrt(t_0) ^ 3.0) ^ 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[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $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

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

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	51840

\[\frac{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w \cdot 0.3333333333333333}}}{{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}^{2}} \]

Alternative 2
Error	0.3
Cost	51712

\[\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt{{\left(e^{0.6666666666666666}\right)}^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

Alternative 3
Error	0.3
Cost	45248

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

Alternative 4
Error	0.3
Cost	19520

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

Alternative 5
Error	0.3
Cost	19456

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

Alternative 6
Error	0.7
Cost	13632

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

Alternative 7
Error	1.3
Cost	13376

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

Alternative 8
Error	1.7
Cost	6592

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

Alternative 9
Error	8.1
Cost	708

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

Alternative 10
Error	13.4
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023067 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))