exp-w crasher

Average Error: 0.3 → 0.3

Time: 14.4s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

(FPCore (w l)
 :precision binary64
 (/
  (pow
   (pow l (exp (* 2.0 (* w 0.3333333333333333))))
   (exp (* w 0.3333333333333333)))
  (exp w)))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

↓

double code(double w, double l) {
	return pow(pow(l, exp((2.0 * (w * 0.3333333333333333)))), exp((w * 0.3333333333333333))) / 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((2.0d0 * (w * 0.3333333333333333d0)))) ** exp((w * 0.3333333333333333d0))) / 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(Math.pow(l, Math.exp((2.0 * (w * 0.3333333333333333)))), Math.exp((w * 0.3333333333333333))) / 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(math.pow(l, math.exp((2.0 * (w * 0.3333333333333333)))), math.exp((w * 0.3333333333333333))) / 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(Float64(2.0 * Float64(w * 0.3333333333333333)))) ^ exp(Float64(w * 0.3333333333333333))) / exp(w))
end

MATLAB

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

↓

function tmp = code(w, l)
	tmp = ((l ^ exp((2.0 * (w * 0.3333333333333333)))) ^ exp((w * 0.3333333333333333))) / 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[N[Power[l, N[Exp[N[(2.0 * N[(w * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Exp[N[(w * 0.3333333333333333), $MachinePrecision]], $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)))): 1 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))): 3 points increase in error, 1 points decrease in error

3. Taylor expanded in l around 0 4.6

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.3

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

6. Applied egg-rr 0.3

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

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	19456

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

Alternative 2
Error	0.9
Cost	13248

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

Alternative 3
Error	0.9
Cost	13184

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

Alternative 4
Error	1.7
Cost	6592

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

Alternative 5
Error	60.8
Cost	64

\[1 \]

Alternative 6
Error	13.6
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2022295 
(FPCore (w l)
  :name "exp-w crasher"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))