exp-w (used to crash)

Average Error: 0.41% → 0.41%

Time: 16.3s

Precision: binary64

Cost: 19584

Math

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

↓

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

FPCore

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

↓

(FPCore (w l) :precision binary64 (* (pow l (exp w)) (/ 1.0 (exp w))))

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)) * (1.0 / exp(w));
}

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)) * (1.0d0 / exp(w))
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)) * (1.0 / Math.exp(w));
}

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)) * (1.0 / math.exp(w))

Julia

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

↓

function code(w, l)
	return Float64((l ^ exp(w)) * Float64(1.0 / exp(w)))
end

MATLAB

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

↓

function tmp = code(w, l)
	tmp = (l ^ exp(w)) * (1.0 / exp(w));
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[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.41

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

2. Simplified 0.41

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

   [Start] 0.41	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   exp-neg [=>] 0.41	\[ \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]
   associate-*l/ [=>] 0.41	\[ \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   *-lft-identity [=>] 0.41	\[ \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Applied egg-rr 7.35

   \[\leadsto \color{blue}{e^{e^{w} \cdot \log \ell - w}} \]

4. Applied egg-rr 0.41

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

5. Final simplification 0.41

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

Alternatives

Alternative 1
Error	0.41%
Cost	19456

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

Alternative 2
Error	1.43%
Cost	13184

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

Alternative 3
Error	2.17%
Cost	12928

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

Alternative 4
Error	2.83%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;w \leq 430:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) + \left(0.5 + w \cdot -0.16666666666666666\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]

Alternative 5
Error	2.78%
Cost	6592

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

Alternative 6
Error	20.76%
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023121 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))