exp-w (used to crash)

Average Error: 0.4% → 0.4%

Time: 16.9s

Precision: binary64

Cost: 32256

Math

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

↓

\[\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{w}\right)\right)} \]

FPCore

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

↓

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (expm1 (log1p (exp w)))))

C

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

↓

double code(double w, double l) {
	return pow(l, exp(w)) / expm1(log1p(exp(w)));
}

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) {
	return Math.pow(l, Math.exp(w)) / Math.expm1(Math.log1p(Math.exp(w)));
}

Python

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

↓

def code(w, l):
	return math.pow(l, math.exp(w)) / math.expm1(math.log1p(math.exp(w)))

Julia

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

↓

function code(w, l)
	return Float64((l ^ exp(w)) / expm1(log1p(exp(w))))
end

Wolfram

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

↓

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(Exp[N[Log[1 + N[Exp[w], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.4

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

2. Simplified 0.4

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

   [Start] 0.4	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   exp-neg [=>] 0.4	\[ \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]
   associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   *-lft-identity [=>] 0.4	\[ \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Applied egg-rr 0.41

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

4. Simplified 0.41

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

   [Start] 0.41	\[ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   associate-*l/ [=>] 0.41	\[ \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
   *-lft-identity [=>] 0.41	\[ \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
   associate-/l/ [=>] 0.41	\[ \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2} \cdot \sqrt[3]{e^{w}}}} \]

5. Applied egg-rr 0.4

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{w}\right)\right)}} \]

6. Final simplification 0.4

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{w}\right)\right)} \]

Alternatives

Alternative 1
Error	0.4%
Cost	19456

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

Alternative 2
Error	1.4%
Cost	13568

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

Alternative 3
Error	1.4%
Cost	13184

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

Alternative 4
Error	2.04%
Cost	12928

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

Alternative 5
Error	2.69%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;w \leq 480:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]

Alternative 6
Error	2.68%
Cost	6592

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

Alternative 7
Error	12.72%
Cost	320

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

Alternative 8
Error	21.5%
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023103 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))