Average Error: 0.3 → 0.3
Time: 13.9s
Precision: binary64
Cost: 51840
\[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
↓
\[\frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\left({\left(\sqrt[3]{e^{w \cdot 0.6666666666666666}}\right)}^{1.5}\right)}^{2}}}{\sqrt[3]{e^{w}}} \]
(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))
↓
(FPCore (w l)
 :precision binary64
 (/
  (/ (pow l (exp w)) (pow (pow (cbrt (exp (* w 0.6666666666666666))) 1.5) 2.0))
  (cbrt (exp w))))
double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}
↓
double code(double w, double l) {
	return (pow(l, exp(w)) / pow(pow(cbrt(exp((w * 0.6666666666666666))), 1.5), 2.0)) / cbrt(exp(w));
}

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.pow(Math.pow(Math.cbrt(Math.exp((w * 0.6666666666666666))), 1.5), 2.0)) / Math.cbrt(Math.exp(w));
}

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end
↓
function code(w, l)
	return Float64(Float64((l ^ exp(w)) / ((cbrt(exp(Float64(w * 0.6666666666666666))) ^ 1.5) ^ 2.0)) / cbrt(exp(w)))
end

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[w_, l_] := N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[N[Power[N[Exp[N[(w * 0.6666666666666666), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

e^{-w} \cdot {\ell}^{\left(e^{w}\right)}
↓
\frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\left({\left(\sqrt[3]{e^{w \cdot 0.6666666666666666}}\right)}^{1.5}\right)}^{2}}}{\sqrt[3]{e^{w}}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
    Proof
    (/.f64 (pow.f64 l (exp.f64 w)) (exp.f64 w)): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (pow.f64 l (exp.f64 w)))) (exp.f64 w)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 w)) (pow.f64 l (exp.f64 w)))): 3 points increase in error, 1 points decrease in error
    (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 w))) (pow.f64 l (exp.f64 w))): 2 points increase in error, 1 points decrease in error

  3. Applied egg-rr0.3

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

  4. Taylor expanded in w around inf 0.3

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

  5. Applied egg-rr0.3

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

  6. Applied egg-rr0.3

    \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\color{blue}{\left({\left(\sqrt[3]{e^{w \cdot 0.6666666666666666}}\right)}^{1.5}\right)}}^{2}}}{\sqrt[3]{e^{w}}} \]

  7. Final simplification0.3

    \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\left({\left(\sqrt[3]{e^{w \cdot 0.6666666666666666}}\right)}^{1.5}\right)}^{2}}}{\sqrt[3]{e^{w}}} \]

Alternatives

Alternative 1
Error0.3
Cost45248
\[\begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{{t_0}^{2}}}{t_0} \end{array} \]
Alternative 2
Error0.3
Cost32512
\[\left(-{\ell}^{\left(e^{w}\right)}\right) \cdot {\left(\frac{-1}{\sqrt[3]{e^{w}}}\right)}^{3} \]
Alternative 3
Error0.3
Cost26240
\[\frac{{\ell}^{\left(e^{w}\right)}}{e^{w \cdot 0.6666666666666666}} \cdot e^{w \cdot -0.3333333333333333} \]
Alternative 4
Error0.3
Cost19456
\[\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]
Alternative 5
Error0.9
Cost13184
\[{\ell}^{\left(e^{w}\right)} \cdot \left(1 - w\right) \]
Alternative 6
Error1.4
Cost12928
\[{\ell}^{\left(e^{w}\right)} \]
Alternative 7
Error1.8
Cost6592
\[\frac{\ell}{e^{w}} \]
Alternative 8
Error12.5
Cost1220
\[\begin{array}{l} \mathbf{if}\;w \leq 8.933743444365022 \cdot 10^{-9}:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(\ell \cdot w\right) \cdot \left(\ell \cdot w\right)}{\ell + \ell \cdot w}\\ \end{array} \]
Alternative 9
Error13.7
Cost64
\[\ell \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (w l)
  :name "exp-w crasher"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))