exp-w crasher

Average Error: 0.3 → 0.3

Time: 13.1s

Precision: binary64

Cost: 19456

Math

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

↓

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

FPCore

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

↓

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (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)) / 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)) / 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)) / 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)) / 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)) / 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)) / 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[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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)))): 2 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))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.8
Cost	13696

\[\begin{array}{l} t_0 := 1 + w \cdot 0.5\\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{t_0}}{t_0} \end{array} \]

Alternative 2
Error	1.3
Cost	13376

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

Alternative 3
Error	1.3
Cost	13376

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

Alternative 4
Error	1.3
Cost	13312

\[\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot 0.5} \]

Alternative 5
Error	1.7
Cost	6660

\[\begin{array}{l} \mathbf{if}\;w \leq 420:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]

Alternative 6
Error	1.7
Cost	6592

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

Alternative 7
Error	12.5
Cost	1220

\[\begin{array}{l} \mathbf{if}\;w \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(\ell \cdot w\right) \cdot \left(\ell \cdot w\right)}{\ell + \ell \cdot w}\\ \end{array} \]

Alternative 8
Error	13.5
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2022329 
(FPCore (w l)
  :name "exp-w crasher"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))