exp-w (used to crash)

Average Accuracy: 71.3% → 72.8%

Time: 19.6s

Precision: binary64

Cost: 51844

Math

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

↓

\[\begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;e^{-w} \cdot t_0 \leq 10^{+307}:\\ \;\;\;\;\frac{t_0}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\sqrt{{\log \left(\sqrt[3]{\ell}\right)}^{2}} \cdot 3}\\ \end{array} \]

FPCore

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

↓

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow l (exp w))))
   (if (<= (* (exp (- w)) t_0) 1e+307)
     (/ t_0 (exp w))
     (exp (* (sqrt (pow (log (cbrt l)) 2.0)) 3.0)))))

C

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

↓

double code(double w, double l) {
	double t_0 = pow(l, exp(w));
	double tmp;
	if ((exp(-w) * t_0) <= 1e+307) {
		tmp = t_0 / exp(w);
	} else {
		tmp = exp((sqrt(pow(log(cbrt(l)), 2.0)) * 3.0));
	}
	return tmp;
}

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.pow(l, Math.exp(w));
	double tmp;
	if ((Math.exp(-w) * t_0) <= 1e+307) {
		tmp = t_0 / Math.exp(w);
	} else {
		tmp = Math.exp((Math.sqrt(Math.pow(Math.log(Math.cbrt(l)), 2.0)) * 3.0));
	}
	return tmp;
}

Julia

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

↓

function code(w, l)
	t_0 = l ^ exp(w)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * t_0) <= 1e+307)
		tmp = Float64(t_0 / exp(w));
	else
		tmp = exp(Float64(sqrt((log(cbrt(l)) ^ 2.0)) * 3.0));
	end
	return tmp
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[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * t$95$0), $MachinePrecision], 1e+307], N[(t$95$0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Sqrt[N[Power[N[Log[N[Power[l, 1/3], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.99999999999999986e306

   1. Initial program 99.6%

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

   2. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
      Step-by-step derivation

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

   if 9.99999999999999986e306 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 0.0%

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

   2. Applied egg-rr 0.0%

      \[\leadsto e^{-w} \cdot \color{blue}{\left({\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}\right)} \]
      Step-by-step derivation

      [Start] 0.0	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
      add-sqr-sqrt [=>] 0.0	\[ e^{-w} \cdot {\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{\left(e^{w}\right)} \]
      unpow-prod-down [=>] 0.0	\[ e^{-w} \cdot \color{blue}{\left({\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}\right)} \]

   3. Simplified 0.0%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}} \]
      Step-by-step derivation

      [Start] 0.0	\[ e^{-w} \cdot \left({\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}\right) \]
      pow-sqr [=>] 0.0	\[ e^{-w} \cdot \color{blue}{{\left(\sqrt{\ell}\right)}^{\left(2 \cdot e^{w}\right)}} \]
      *-commutative [=>] 0.0	\[ e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\color{blue}{\left(e^{w} \cdot 2\right)}} \]

   4. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\ell}\right) \cdot 3}} \]
      Step-by-step derivation

      [Start] 0.0	\[ e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)} \]
      add-sqr-sqrt [=>] 0.0	\[ \color{blue}{\sqrt{e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}} \cdot \sqrt{e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}} \]
      sqrt-unprod [=>] 0.0	\[ \color{blue}{\sqrt{\left(e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right) \cdot \left(e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}} \]
      pow2 [=>] 0.0	\[ \sqrt{\color{blue}{{\left(e^{-w} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}}} \]
      add-sqr-sqrt [=>] 0.0	\[ \sqrt{{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}} \]
      sqrt-unprod [=>] 0.0	\[ \sqrt{{\left(e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}} \]
      sqr-neg [=>] 0.0	\[ \sqrt{{\left(e^{\sqrt{\color{blue}{w \cdot w}}} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}} \]
      sqrt-unprod [<=] 0.0	\[ \sqrt{{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}} \]
      add-sqr-sqrt [<=] 1.6	\[ \sqrt{{\left(e^{\color{blue}{w}} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}} \]
      pow-prod-down [<=] 1.6	\[ \sqrt{\color{blue}{{\left(e^{w}\right)}^{2} \cdot {\left({\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}}} \]
      pow2 [<=] 1.6	\[ \sqrt{\color{blue}{\left(e^{w} \cdot e^{w}\right)} \cdot {\left({\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}^{2}} \]
      pow2 [<=] 1.6	\[ \sqrt{\left(e^{w} \cdot e^{w}\right) \cdot \color{blue}{\left({\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}\right)}} \]

   5. Applied egg-rr 5.2%

      \[\leadsto e^{\color{blue}{\sqrt{{\log \left(\sqrt[3]{\ell}\right)}^{2}}} \cdot 3} \]
      Step-by-step derivation

      [Start] 3.7	\[ e^{\log \left(\sqrt[3]{\ell}\right) \cdot 3} \]
      add-sqr-sqrt [=>] 2.5	\[ e^{\color{blue}{\left(\sqrt{\log \left(\sqrt[3]{\ell}\right)} \cdot \sqrt{\log \left(\sqrt[3]{\ell}\right)}\right)} \cdot 3} \]
      sqrt-unprod [=>] 5.2	\[ e^{\color{blue}{\sqrt{\log \left(\sqrt[3]{\ell}\right) \cdot \log \left(\sqrt[3]{\ell}\right)}} \cdot 3} \]
      pow2 [=>] 5.2	\[ e^{\sqrt{\color{blue}{{\log \left(\sqrt[3]{\ell}\right)}^{2}}} \cdot 3} \]

3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{+307}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\sqrt{{\log \left(\sqrt[3]{\ell}\right)}^{2}} \cdot 3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.8%
Cost	45316

\[\begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;e^{-w} \cdot t_0 \leq 10^{+307}:\\ \;\;\;\;\frac{t_0}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\sqrt{{\log \ell}^{2}}}\\ \end{array} \]

Alternative 2
Accuracy	71.3%
Cost	19456

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

Alternative 3
Accuracy	70.4%
Cost	13376

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

Alternative 4
Accuracy	69.8%
Cost	6592

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

Alternative 5
Accuracy	70.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;w \leq 0.00145:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\ell + -1\right)\\ \end{array} \]

Alternative 6
Accuracy	57.5%
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023157 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))