exp-w (used to crash)

Average Accuracy: 99.6% → 99.6%

Time: 19.2s

Precision: binary64

Cost: 109888

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ t_1 := \sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{t_0}\\ \frac{1}{{\left(t_1 \cdot t_1\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{t_0} \end{array} \]

FPCore

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

↓

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (exp w)))
        (t_1 (* (cbrt (pow (exp w) 0.16666666666666666)) (cbrt t_0))))
   (* (/ 1.0 (pow (* t_1 t_1) 2.0)) (/ (pow l (exp w)) t_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));
	double t_1 = cbrt(pow(exp(w), 0.16666666666666666)) * cbrt(t_0);
	return (1.0 / pow((t_1 * t_1), 2.0)) * (pow(l, exp(w)) / t_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));
	double t_1 = Math.cbrt(Math.pow(Math.exp(w), 0.16666666666666666)) * Math.cbrt(t_0);
	return (1.0 / Math.pow((t_1 * t_1), 2.0)) * (Math.pow(l, Math.exp(w)) / t_0);
}

Julia

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

↓

function code(w, l)
	t_0 = cbrt(exp(w))
	t_1 = Float64(cbrt((exp(w) ^ 0.16666666666666666)) * cbrt(t_0))
	return Float64(Float64(1.0 / (Float64(t_1 * t_1) ^ 2.0)) * Float64((l ^ exp(w)) / t_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]}, Block[{t$95$1 = N[(N[Power[N[Power[N[Exp[w], $MachinePrecision], 0.16666666666666666], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / t$95$0), $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. Applied egg-rr 99.6%

   \[\leadsto \frac{1}{{\color{blue}{\left(\left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   Proof

   [Start] 99.6	\[ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   add-cube-cbrt [=>] 99.6	\[ \frac{1}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)}}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   *-commutative [=>] 99.6	\[ \frac{1}{{\color{blue}{\left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   add-sqr-sqrt [=>] 99.6	\[ \frac{1}{{\left(\sqrt[3]{\color{blue}{\sqrt{\sqrt[3]{e^{w}}} \cdot \sqrt{\sqrt[3]{e^{w}}}}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   cbrt-prod [=>] 99.6	\[ \frac{1}{{\left(\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   unswap-sqr [=>] 99.6	\[ \frac{1}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   pow1/3 [=>] 99.6	\[ \frac{1}{{\left(\left(\sqrt[3]{\sqrt{\color{blue}{{\left(e^{w}\right)}^{0.3333333333333333}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   sqrt-pow1 [=>] 99.6	\[ \frac{1}{{\left(\left(\sqrt[3]{\color{blue}{{\left(e^{w}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   metadata-eval [=>] 99.6	\[ \frac{1}{{\left(\left(\sqrt[3]{{\left(e^{w}\right)}^{\color{blue}{0.16666666666666666}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{e^{w}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   pow1/3 [=>] 99.6	\[ \frac{1}{{\left(\left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{\sqrt{\color{blue}{{\left(e^{w}\right)}^{0.3333333333333333}}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   sqrt-pow1 [=>] 99.6	\[ \frac{1}{{\left(\left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{\color{blue}{{\left(e^{w}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   metadata-eval [=>] 99.6	\[ \frac{1}{{\left(\left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{{\left(e^{w}\right)}^{\color{blue}{0.16666666666666666}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

5. Final simplification 99.6%

   \[\leadsto \frac{1}{{\left(\left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \left(\sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	109760

\[\begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ t_1 := \sqrt[3]{{\left(e^{w}\right)}^{0.16666666666666666}} \cdot \sqrt[3]{t_0}\\ \frac{{\ell}^{\left(e^{w}\right)}}{t_0 \cdot {\left(t_1 \cdot t_1\right)}^{2}} \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	58176

\[\begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{{\ell}^{\left(e^{w}\right)}}{t_0} \cdot \frac{1}{{\log \left(e^{t_0}\right)}^{2}} \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	45376

\[\begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{{\ell}^{\left(e^{w}\right)}}{t_0} \cdot \frac{1}{{t_0}^{2}} \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	45248

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

Alternative 5
Accuracy	99.6%
Cost	39104

\[\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \left({\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}\right) \]

Alternative 6
Accuracy	99.6%
Cost	38976

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

Alternative 7
Accuracy	99.6%
Cost	32512

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

Alternative 8
Accuracy	99.6%
Cost	19456

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

Alternative 9
Accuracy	98.7%
Cost	13696

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

Alternative 10
Accuracy	97.9%
Cost	13376

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

Alternative 11
Accuracy	97.4%
Cost	6592

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

Alternative 12
Accuracy	86.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;w \leq 0.000102:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \ell \cdot \left(1 - w\right)\right) + -1\\ \end{array} \]

Alternative 13
Accuracy	78.3%
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023135 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))