exp-w (used to crash)

Average Error: 0.2 → 0.2

Time: 16.1s

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 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   exp-neg [=>] 0.2	\[ \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]
   associate-*l/ [=>] 0.2	\[ \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   *-lft-identity [=>] 0.2	\[ \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Applied egg-rr 0.2

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

4. Simplified 0.2

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

   [Start] 0.2	\[ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   associate-*l/ [=>] 0.2	\[ \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
   *-lft-identity [=>] 0.2	\[ \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
   associate-/l/ [=>] 0.2	\[ \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2} \cdot \sqrt[3]{e^{w}}}} \]

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	32384

\[\frac{{\ell}^{\left(e^{w}\right)}}{\log \left(1 + \mathsf{expm1}\left(e^{w}\right)\right)} \]

Alternative 2
Error	0.2
Cost	19520

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

Alternative 3
Error	0.2
Cost	19456

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

Alternative 4
Error	0.7
Cost	13568

\[\frac{{\ell}^{\left(e^{w}\right)}}{w + \left(1 + 0.5 \cdot \left(w \cdot w\right)\right)} \]

Alternative 5
Error	0.8
Cost	13184

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

Alternative 6
Error	1.0
Cost	6912

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

Alternative 7
Error	1.3
Cost	6656

\[{\ell}^{\left(w + 1\right)} \]

Alternative 8
Error	1.6
Cost	6592

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

Alternative 9
Error	1.6
Cost	708

\[\begin{array}{l} t_0 := \frac{\ell}{w + 1}\\ \mathbf{if}\;w \leq 0.102:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \end{array} \]

Alternative 10
Error	7.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;w \leq 1:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \]

Alternative 11
Error	7.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;w \leq 0.85:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \]

Alternative 12
Error	7.6
Cost	320

\[\frac{\ell}{w + 1} \]

Alternative 13
Error	13.7
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023054 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))