exp-w (used to crash)

Average Accuracy: 99.6% → 99.4%

Time: 19.8s

Precision: binary64

Cost: 39168

Math

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

↓

\[\frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} + -1\right) \cdot e^{w \cdot 0.3333333333333333}} \]

FPCore

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

↓

(FPCore (w l)
 :precision binary64
 (/
  (pow l (exp w))
  (*
   (+ (exp (log1p (exp (* w 0.6666666666666666)))) -1.0)
   (exp (* w 0.3333333333333333)))))

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)) / ((exp(log1p(exp((w * 0.6666666666666666)))) + -1.0) * exp((w * 0.3333333333333333)));
}

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.exp(Math.log1p(Math.exp((w * 0.6666666666666666)))) + -1.0) * Math.exp((w * 0.3333333333333333)));
}

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.exp(math.log1p(math.exp((w * 0.6666666666666666)))) + -1.0) * math.exp((w * 0.3333333333333333)))

Julia

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

↓

function code(w, l)
	return Float64((l ^ exp(w)) / Float64(Float64(exp(log1p(exp(Float64(w * 0.6666666666666666)))) + -1.0) * exp(Float64(w * 0.3333333333333333))))
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[(N[(N[Exp[N[Log[1 + N[Exp[N[(w * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(w * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Simplified 99.6%

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

   [Start] 99.6	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   exp-neg [=>] 99.6	\[ \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]
   associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   *-lft-identity [=>] 99.6	\[ \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Applied egg-rr 99.6%

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

   [Start] 99.6	\[ \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]
   *-un-lft-identity [=>] 99.6	\[ \frac{\color{blue}{1 \cdot {\ell}^{\left(e^{w}\right)}}}{e^{w}} \]
   add-cube-cbrt [=>] 99.6	\[ \frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]
   times-frac [=>] 99.6	\[ \color{blue}{\frac{1}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
   pow2 [=>] 99.6	\[ \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

4. Simplified 99.6%

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

   [Start] 99.6	\[ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
   *-lft-identity [=>] 99.6	\[ \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
   associate-/l/ [=>] 99.6	\[ \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2} \cdot \sqrt[3]{e^{w}}}} \]

5. Applied egg-rr 99.4%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} - 1\right)} \cdot \sqrt[3]{e^{w}}} \]
   Proof

   [Start] 99.6	\[ \frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2} \cdot \sqrt[3]{e^{w}}} \]
   expm1-log1p-u [=>] 99.6	\[ \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{e^{w}}\right)}^{2}\right)\right)} \cdot \sqrt[3]{e^{w}}} \]
   expm1-udef [=>] 99.4	\[ \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{e^{w}}\right)}^{2}\right)} - 1\right)} \cdot \sqrt[3]{e^{w}}} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} - 1\right) \cdot \color{blue}{e^{w \cdot 0.3333333333333333}}} \]
   Proof

   [Start] 99.4	\[ \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} - 1\right) \cdot \sqrt[3]{e^{w}}} \]
   pow1/3 [=>] 99.4	\[ \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} - 1\right) \cdot \color{blue}{{\left(e^{w}\right)}^{0.3333333333333333}}} \]
   pow-exp [=>] 99.4	\[ \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} - 1\right) \cdot \color{blue}{e^{w \cdot 0.3333333333333333}}} \]

7. Final simplification 99.4%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{\mathsf{log1p}\left(e^{w \cdot 0.6666666666666666}\right)} + -1\right) \cdot e^{w \cdot 0.3333333333333333}} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	19520

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

Alternative 2
Accuracy	99.6%
Cost	19456

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

Alternative 3
Accuracy	98.0%
Cost	13376

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

Alternative 4
Accuracy	98.0%
Cost	13376

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

Alternative 5
Accuracy	97.2%
Cost	6592

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

Alternative 6
Accuracy	79.7%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell \cdot \left(1 - w\right) + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{elif}\;w \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(\ell \cdot w\right) \cdot \left(\ell \cdot w\right)}{\ell + \ell \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 7
Accuracy	78.2%
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023136 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))