exp-w (used to crash)

Average Error: 0.44% → 0.45%

Time: 16.2s

Precision: binary64

Cost: 26240

Math

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

↓

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

FPCore

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

↓

(FPCore (w l)
 :precision binary64
 (/
  (/ (pow l (exp w)) (exp (* w 0.3333333333333333)))
  (exp (* w 0.6666666666666666))))

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((w * 0.3333333333333333))) / exp((w * 0.6666666666666666));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

↓

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l ** exp(w)) / exp((w * 0.3333333333333333d0))) / exp((w * 0.6666666666666666d0))
end function

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((w * 0.3333333333333333))) / Math.exp((w * 0.6666666666666666));
}

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((w * 0.3333333333333333))) / math.exp((w * 0.6666666666666666))

Julia

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

↓

function code(w, l)
	return Float64(Float64((l ^ exp(w)) / exp(Float64(w * 0.3333333333333333))) / exp(Float64(w * 0.6666666666666666)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

↓

function tmp = code(w, l)
	tmp = ((l ^ exp(w)) / exp((w * 0.3333333333333333))) / exp((w * 0.6666666666666666));
end

Wolfram

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[Exp[N[(w * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(w * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.44

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

2. Simplified 0.44

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

   [Start] 0.44	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   exp-neg [=>] 0.44	\[ \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]
   associate-*l/ [=>] 0.44	\[ \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   *-lft-identity [=>] 0.44	\[ \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Applied egg-rr 0.44

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

4. Simplified 0.44

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

   [Start] 0.44	\[ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]
   associate-*l/ [=>] 0.44	\[ \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
   *-lft-identity [=>] 0.44	\[ \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

5. Applied egg-rr 0.45

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

6. Applied egg-rr 0.45

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

7. Final simplification 0.45

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

Alternatives

Alternative 1
Error	0.44%
Cost	19520

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

Alternative 2
Error	0.44%
Cost	19456

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

Alternative 3
Error	1.94%
Cost	13440

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

Alternative 4
Error	1.94%
Cost	13376

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

Alternative 5
Error	2.78%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;w \leq 460:\\ \;\;\;\;\ell \cdot \left(1 - w\right) + \left(w \cdot w\right) \cdot \left(\ell \cdot 0.5 + w \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]

Alternative 6
Error	2.73%
Cost	6656

\[\ell \cdot e^{-w} \]

Alternative 7
Error	2.73%
Cost	6592

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

Alternative 8
Error	13.26%
Cost	1348

\[\begin{array}{l} t_0 := \ell \cdot \left(1 - w\right)\\ \mathbf{if}\;w \leq 0.033:\\ \;\;\;\;t_0 + \left(w \cdot w\right) \cdot \left(\ell \cdot 0.5 + w \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \end{array} \]

Alternative 9
Error	13.26%
Cost	708

\[\begin{array}{l} t_0 := \ell \cdot \left(1 - w\right)\\ \mathbf{if}\;w \leq 0.033:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \end{array} \]

Alternative 10
Error	21.22%
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023104 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))