exp-w (used to crash)

Percentage Accurate: 99.3% → 99.7%

Time: 18.6s

Alternatives: 13

Speedup: 0.5×

• could not determine a ground truth

  1. w = 2.00000000000000007e154
  2. l = -2

Specification

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

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

C

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

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

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

C

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

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;t_0 \cdot e^{-w} \leq 10^{+305}:\\ \;\;\;\;\frac{t_0}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow l (exp w))))
   (if (<= (* t_0 (exp (- w))) 1e+305)
     (/ t_0 (exp w))
     (exp (- (* (exp w) (log l)) w)))))

C

double code(double w, double l) {
	double t_0 = pow(l, exp(w));
	double tmp;
	if ((t_0 * exp(-w)) <= 1e+305) {
		tmp = t_0 / exp(w);
	} else {
		tmp = exp(((exp(w) * log(l)) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l ** exp(w)
    if ((t_0 * exp(-w)) <= 1d+305) then
        tmp = t_0 / exp(w)
    else
        tmp = exp(((exp(w) * log(l)) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.pow(l, Math.exp(w));
	double tmp;
	if ((t_0 * Math.exp(-w)) <= 1e+305) {
		tmp = t_0 / Math.exp(w);
	} else {
		tmp = Math.exp(((Math.exp(w) * Math.log(l)) - w));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.pow(l, math.exp(w))
	tmp = 0
	if (t_0 * math.exp(-w)) <= 1e+305:
		tmp = t_0 / math.exp(w)
	else:
		tmp = math.exp(((math.exp(w) * math.log(l)) - w))
	return tmp

Julia

function code(w, l)
	t_0 = l ^ exp(w)
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-w))) <= 1e+305)
		tmp = Float64(t_0 / exp(w));
	else
		tmp = exp(Float64(Float64(exp(w) * log(l)) - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = l ^ exp(w);
	tmp = 0.0;
	if ((t_0 * exp(-w)) <= 1e+305)
		tmp = t_0 / exp(w);
	else
		tmp = exp(((exp(w) * log(l)) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Exp[(-w)], $MachinePrecision]), $MachinePrecision], 1e+305], N[(t$95$0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Exp[w], $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e304

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

   3. Simplified 99.7%

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

   if 9.9999999999999994e304 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 93.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-lft-identity 93.2%

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

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-exp-log 93.2%

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

      2. log-div 93.2%

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

      3. log-pow 93.2%

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

      4. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{e^{w} \cdot \log \ell - w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 10^{+305}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{\sqrt[3]{{\left(e^{w}\right)}^{2}}} \end{array} \]

FPCore

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

C

double code(double w, double l) {
	return (pow(l, exp(w)) / cbrt(exp(w))) / cbrt(pow(exp(w), 2.0));
}

Java

public static double code(double w, double l) {
	return (Math.pow(l, Math.exp(w)) / Math.cbrt(Math.exp(w))) / Math.cbrt(Math.pow(Math.exp(w), 2.0));
}

Julia

function code(w, l)
	return Float64(Float64((l ^ exp(w)) / cbrt(exp(w))) / cbrt((exp(w) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 98.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 98.2%

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

   2. add-cube-cbrt 98.2%

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

   3. times-frac 98.2%

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

   4. cbrt-unprod 98.2%

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

   5. prod-exp 98.2%

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

5. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{w + w}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
6. Step-by-step derivation

   1. associate-*l/ 98.2%

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

   2. *-lft-identity 98.2%

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

   3. count-2 98.2%

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

   4. *-commutative 98.2%

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

   5. exp-prod 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.0036:\\ \;\;\;\;\frac{e^{\log \ell}}{e^{w}}\\ \mathbf{elif}\;w \leq 0.045:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{elif}\;w \leq 10^{+18}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.0036)
   (/ (exp (log l)) (exp w))
   (if (<= w 0.045)
     (+ l (* w (- (* l (log l)) l)))
     (if (<= w 1e+18) (log (exp l)) (/ (* (log l) (* l w)) (exp w))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.0036) {
		tmp = exp(log(l)) / exp(w);
	} else if (w <= 0.045) {
		tmp = l + (w * ((l * log(l)) - l));
	} else if (w <= 1e+18) {
		tmp = log(exp(l));
	} else {
		tmp = (log(l) * (l * w)) / exp(w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.0036d0)) then
        tmp = exp(log(l)) / exp(w)
    else if (w <= 0.045d0) then
        tmp = l + (w * ((l * log(l)) - l))
    else if (w <= 1d+18) then
        tmp = log(exp(l))
    else
        tmp = (log(l) * (l * w)) / exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.0036) {
		tmp = Math.exp(Math.log(l)) / Math.exp(w);
	} else if (w <= 0.045) {
		tmp = l + (w * ((l * Math.log(l)) - l));
	} else if (w <= 1e+18) {
		tmp = Math.log(Math.exp(l));
	} else {
		tmp = (Math.log(l) * (l * w)) / Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.0036:
		tmp = math.exp(math.log(l)) / math.exp(w)
	elif w <= 0.045:
		tmp = l + (w * ((l * math.log(l)) - l))
	elif w <= 1e+18:
		tmp = math.log(math.exp(l))
	else:
		tmp = (math.log(l) * (l * w)) / math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.0036)
		tmp = Float64(exp(log(l)) / exp(w));
	elseif (w <= 0.045)
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	elseif (w <= 1e+18)
		tmp = log(exp(l));
	else
		tmp = Float64(Float64(log(l) * Float64(l * w)) / exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.0036)
		tmp = exp(log(l)) / exp(w);
	elseif (w <= 0.045)
		tmp = l + (w * ((l * log(l)) - l));
	elseif (w <= 1e+18)
		tmp = log(exp(l));
	else
		tmp = (log(l) * (l * w)) / exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.0036], N[(N[Exp[N[Log[l], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.045], N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1e+18], N[Log[N[Exp[l], $MachinePrecision]], $MachinePrecision], N[(N[(N[Log[l], $MachinePrecision] * N[(l * w), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if w < -0.0035999999999999999

   1. Initial program 99.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-exp-log 99.9%

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

      2. log-pow 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 96.8%

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

   if -0.0035999999999999999 < w < 0.044999999999999998

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 99.2%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

   if 0.044999999999999998 < w < 1e18

   1. Initial program 33.3%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 33.3%

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

      2. associate-*l/ 33.3%

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

      3. *-lft-identity 33.3%

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

   3. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 33.3%

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

      2. pow3 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in w around 0 8.2%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 8.2%

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

      2. rem-cube-cbrt 8.2%

         \[\leadsto \color{blue}{\ell} \]

      3. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   if 1e18 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 100.0%

      \[\leadsto \frac{\color{blue}{\ell + \ell \cdot \left(w \cdot \log \ell\right)}}{e^{w}} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in w around inf 100.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.0036:\\ \;\;\;\;\frac{e^{\log \ell}}{e^{w}}\\ \mathbf{elif}\;w \leq 0.045:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{elif}\;w \leq 10^{+18}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \end{array} \]

FPCore

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

C

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 = (l ** exp(w)) / exp(w)
end function

Java

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.pow(l, math.exp(w)) / math.exp(w)

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 5: 83.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(e^{\ell}\right)\\ t_1 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell}\right)\right)\\ \mathbf{if}\;w \leq -9.2 \cdot 10^{+266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -1 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -0.64:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \leq 0.075:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (log (exp l))) (t_1 (log1p (expm1 (/ 1.0 l)))))
   (if (<= w -9.2e+266)
     t_0
     (if (<= w -1e+35)
       t_1
       (if (<= w -1.8e+20)
         t_0
         (if (<= w -0.64)
           t_1
           (if (<= w 0.075) (+ l (* w (- (* l (log l)) l))) t_0)))))))

C

double code(double w, double l) {
	double t_0 = log(exp(l));
	double t_1 = log1p(expm1((1.0 / l)));
	double tmp;
	if (w <= -9.2e+266) {
		tmp = t_0;
	} else if (w <= -1e+35) {
		tmp = t_1;
	} else if (w <= -1.8e+20) {
		tmp = t_0;
	} else if (w <= -0.64) {
		tmp = t_1;
	} else if (w <= 0.075) {
		tmp = l + (w * ((l * log(l)) - l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.log(Math.exp(l));
	double t_1 = Math.log1p(Math.expm1((1.0 / l)));
	double tmp;
	if (w <= -9.2e+266) {
		tmp = t_0;
	} else if (w <= -1e+35) {
		tmp = t_1;
	} else if (w <= -1.8e+20) {
		tmp = t_0;
	} else if (w <= -0.64) {
		tmp = t_1;
	} else if (w <= 0.075) {
		tmp = l + (w * ((l * Math.log(l)) - l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.log(math.exp(l))
	t_1 = math.log1p(math.expm1((1.0 / l)))
	tmp = 0
	if w <= -9.2e+266:
		tmp = t_0
	elif w <= -1e+35:
		tmp = t_1
	elif w <= -1.8e+20:
		tmp = t_0
	elif w <= -0.64:
		tmp = t_1
	elif w <= 0.075:
		tmp = l + (w * ((l * math.log(l)) - l))
	else:
		tmp = t_0
	return tmp

Julia

function code(w, l)
	t_0 = log(exp(l))
	t_1 = log1p(expm1(Float64(1.0 / l)))
	tmp = 0.0
	if (w <= -9.2e+266)
		tmp = t_0;
	elseif (w <= -1e+35)
		tmp = t_1;
	elseif (w <= -1.8e+20)
		tmp = t_0;
	elseif (w <= -0.64)
		tmp = t_1;
	elseif (w <= 0.075)
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Log[N[Exp[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[1 + N[(Exp[N[(1.0 / l), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[w, -9.2e+266], t$95$0, If[LessEqual[w, -1e+35], t$95$1, If[LessEqual[w, -1.8e+20], t$95$0, If[LessEqual[w, -0.64], t$95$1, If[LessEqual[w, 0.075], N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -9.2000000000000004e266 or -9.9999999999999997e34 < w < -1.8e20 or 0.0749999999999999972 < w

   1. Initial program 93.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-lft-identity 93.2%

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

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 93.2%

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

      2. pow3 93.2%

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

   5. Applied egg-rr 93.2%

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

   6. Taylor expanded in w around 0 5.3%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 5.3%

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

      2. rem-cube-cbrt 5.3%

         \[\leadsto \color{blue}{\ell} \]

      3. add-log-exp 91.7%

         \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   8. Applied egg-rr 91.7%

      \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   if -9.2000000000000004e266 < w < -9.9999999999999997e34 or -1.8e20 < w < -0.640000000000000013

   1. Initial program 99.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 3.2%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 3.2%

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

      2. rem-cube-cbrt 3.2%

         \[\leadsto \color{blue}{\ell} \]

      3. add-exp-log 3.2%

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

      4. add-sqr-sqrt 1.6%

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

      5. sqrt-unprod 5.5%

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

      6. sqr-neg 5.5%

         \[\leadsto e^{\sqrt{\color{blue}{\left(-\log \ell\right) \cdot \left(-\log \ell\right)}}} \]

      7. sqrt-unprod 3.9%

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

      8. add-sqr-sqrt 4.8%

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

      9. rec-exp 4.8%

         \[\leadsto \color{blue}{\frac{1}{e^{\log \ell}}} \]

      10. add-exp-log 4.8%

          \[\leadsto \frac{1}{\color{blue}{\ell}} \]

      11. add-sqr-sqrt 4.8%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      12. associate-/r* 4.8%

          \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]

   8. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 69.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}\right)\right)} \]

      2. associate-/l/ 69.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)\right) \]

      3. add-sqr-sqrt 69.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\color{blue}{\ell}}\right)\right) \]

   10. Applied egg-rr 69.3%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell}\right)\right)} \]

   if -0.640000000000000013 < w < 0.0749999999999999972

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 99.2%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -9.2 \cdot 10^{+266}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{elif}\;w \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell}\right)\right)\\ \mathbf{elif}\;w \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{elif}\;w \leq -0.64:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\ell}\right)\right)\\ \mathbf{elif}\;w \leq 0.075:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \end{array} \]

Alternative 6: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(e^{\ell}\right)\\ \mathbf{if}\;w \leq -3.1 \cdot 10^{+262}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -2.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{\log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -0.58 \lor \neg \left(w \leq 0.028\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (log (exp l))))
   (if (<= w -3.1e+262)
     t_0
     (if (<= w -2.05e+35)
       (/ (* (log l) (* l w)) (exp w))
       (if (or (<= w -0.58) (not (<= w 0.028)))
         t_0
         (+ l (* w (- (* l (log l)) l))))))))

C

double code(double w, double l) {
	double t_0 = log(exp(l));
	double tmp;
	if (w <= -3.1e+262) {
		tmp = t_0;
	} else if (w <= -2.05e+35) {
		tmp = (log(l) * (l * w)) / exp(w);
	} else if ((w <= -0.58) || !(w <= 0.028)) {
		tmp = t_0;
	} else {
		tmp = l + (w * ((l * log(l)) - l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(exp(l))
    if (w <= (-3.1d+262)) then
        tmp = t_0
    else if (w <= (-2.05d+35)) then
        tmp = (log(l) * (l * w)) / exp(w)
    else if ((w <= (-0.58d0)) .or. (.not. (w <= 0.028d0))) then
        tmp = t_0
    else
        tmp = l + (w * ((l * log(l)) - l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.log(Math.exp(l));
	double tmp;
	if (w <= -3.1e+262) {
		tmp = t_0;
	} else if (w <= -2.05e+35) {
		tmp = (Math.log(l) * (l * w)) / Math.exp(w);
	} else if ((w <= -0.58) || !(w <= 0.028)) {
		tmp = t_0;
	} else {
		tmp = l + (w * ((l * Math.log(l)) - l));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.log(math.exp(l))
	tmp = 0
	if w <= -3.1e+262:
		tmp = t_0
	elif w <= -2.05e+35:
		tmp = (math.log(l) * (l * w)) / math.exp(w)
	elif (w <= -0.58) or not (w <= 0.028):
		tmp = t_0
	else:
		tmp = l + (w * ((l * math.log(l)) - l))
	return tmp

Julia

function code(w, l)
	t_0 = log(exp(l))
	tmp = 0.0
	if (w <= -3.1e+262)
		tmp = t_0;
	elseif (w <= -2.05e+35)
		tmp = Float64(Float64(log(l) * Float64(l * w)) / exp(w));
	elseif ((w <= -0.58) || !(w <= 0.028))
		tmp = t_0;
	else
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = log(exp(l));
	tmp = 0.0;
	if (w <= -3.1e+262)
		tmp = t_0;
	elseif (w <= -2.05e+35)
		tmp = (log(l) * (l * w)) / exp(w);
	elseif ((w <= -0.58) || ~((w <= 0.028)))
		tmp = t_0;
	else
		tmp = l + (w * ((l * log(l)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Log[N[Exp[l], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[w, -3.1e+262], t$95$0, If[LessEqual[w, -2.05e+35], N[(N[(N[Log[l], $MachinePrecision] * N[(l * w), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[w, -0.58], N[Not[LessEqual[w, 0.028]], $MachinePrecision]], t$95$0, N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -3.09999999999999991e262 or -2.0499999999999999e35 < w < -0.57999999999999996 or 0.0280000000000000006 < w

   1. Initial program 93.5%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 93.5%

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

      2. associate-*l/ 93.5%

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

      3. *-lft-identity 93.5%

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

   3. Simplified 93.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 93.5%

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

      2. pow3 93.5%

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

   5. Applied egg-rr 93.5%

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

   6. Taylor expanded in w around 0 5.4%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 5.4%

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

      2. rem-cube-cbrt 5.4%

         \[\leadsto \color{blue}{\ell} \]

      3. add-log-exp 87.4%

         \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   8. Applied egg-rr 87.4%

      \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   if -3.09999999999999991e262 < w < -2.0499999999999999e35

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 73.7%

      \[\leadsto \frac{\color{blue}{\ell + \ell \cdot \left(w \cdot \log \ell\right)}}{e^{w}} \]
   5. Step-by-step derivation

      1. associate-*r* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in w around inf 73.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}} \]
   8. Step-by-step derivation

      1. associate-*r* 73.7%

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

   9. Simplified 73.7%

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

   if -0.57999999999999996 < w < 0.0280000000000000006

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 99.2%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.1 \cdot 10^{+262}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{elif}\;w \leq -2.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{\log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -0.58 \lor \neg \left(w \leq 0.028\right):\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \]

Alternative 7: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{if}\;w \leq -4.5 \cdot 10^{+262}:\\ \;\;\;\;\ell + \left|t_0\right|\\ \mathbf{elif}\;w \leq -8.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -0.35 \lor \neg \left(w \leq 0.04\right):\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (- (* l (log l)) l))))
   (if (<= w -4.5e+262)
     (+ l (fabs t_0))
     (if (<= w -8.1e+34)
       (/ (* (log l) (* l w)) (exp w))
       (if (or (<= w -0.35) (not (<= w 0.04))) (log (exp l)) (+ l t_0))))))

C

double code(double w, double l) {
	double t_0 = w * ((l * log(l)) - l);
	double tmp;
	if (w <= -4.5e+262) {
		tmp = l + fabs(t_0);
	} else if (w <= -8.1e+34) {
		tmp = (log(l) * (l * w)) / exp(w);
	} else if ((w <= -0.35) || !(w <= 0.04)) {
		tmp = log(exp(l));
	} else {
		tmp = l + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w * ((l * log(l)) - l)
    if (w <= (-4.5d+262)) then
        tmp = l + abs(t_0)
    else if (w <= (-8.1d+34)) then
        tmp = (log(l) * (l * w)) / exp(w)
    else if ((w <= (-0.35d0)) .or. (.not. (w <= 0.04d0))) then
        tmp = log(exp(l))
    else
        tmp = l + t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = w * ((l * Math.log(l)) - l);
	double tmp;
	if (w <= -4.5e+262) {
		tmp = l + Math.abs(t_0);
	} else if (w <= -8.1e+34) {
		tmp = (Math.log(l) * (l * w)) / Math.exp(w);
	} else if ((w <= -0.35) || !(w <= 0.04)) {
		tmp = Math.log(Math.exp(l));
	} else {
		tmp = l + t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = w * ((l * math.log(l)) - l)
	tmp = 0
	if w <= -4.5e+262:
		tmp = l + math.fabs(t_0)
	elif w <= -8.1e+34:
		tmp = (math.log(l) * (l * w)) / math.exp(w)
	elif (w <= -0.35) or not (w <= 0.04):
		tmp = math.log(math.exp(l))
	else:
		tmp = l + t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(w * Float64(Float64(l * log(l)) - l))
	tmp = 0.0
	if (w <= -4.5e+262)
		tmp = Float64(l + abs(t_0));
	elseif (w <= -8.1e+34)
		tmp = Float64(Float64(log(l) * Float64(l * w)) / exp(w));
	elseif ((w <= -0.35) || !(w <= 0.04))
		tmp = log(exp(l));
	else
		tmp = Float64(l + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = w * ((l * log(l)) - l);
	tmp = 0.0;
	if (w <= -4.5e+262)
		tmp = l + abs(t_0);
	elseif (w <= -8.1e+34)
		tmp = (log(l) * (l * w)) / exp(w);
	elseif ((w <= -0.35) || ~((w <= 0.04)))
		tmp = log(exp(l));
	else
		tmp = l + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4.5e+262], N[(l + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -8.1e+34], N[(N[(N[Log[l], $MachinePrecision] * N[(l * w), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[w, -0.35], N[Not[LessEqual[w, 0.04]], $MachinePrecision]], N[Log[N[Exp[l], $MachinePrecision]], $MachinePrecision], N[(l + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if w < -4.49999999999999972e262

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 1.4%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 1.4%

         \[\leadsto \ell + \color{blue}{\sqrt{w \cdot \left(\ell \cdot \log \ell - \ell\right)} \cdot \sqrt{w \cdot \left(\ell \cdot \log \ell - \ell\right)}} \]

      2. sqrt-unprod 89.3%

         \[\leadsto \ell + \color{blue}{\sqrt{\left(w \cdot \left(\ell \cdot \log \ell - \ell\right)\right) \cdot \left(w \cdot \left(\ell \cdot \log \ell - \ell\right)\right)}} \]

      3. pow2 89.3%

         \[\leadsto \ell + \sqrt{\color{blue}{{\left(w \cdot \left(\ell \cdot \log \ell - \ell\right)\right)}^{2}}} \]

      4. add-log-exp 89.3%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \left(\color{blue}{\log \left(e^{\ell \cdot \log \ell}\right)} - \ell\right)\right)}^{2}} \]

      5. *-commutative 89.3%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \left(\log \left(e^{\color{blue}{\log \ell \cdot \ell}}\right) - \ell\right)\right)}^{2}} \]

      6. exp-to-pow 89.3%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \left(\log \color{blue}{\left({\ell}^{\ell}\right)} - \ell\right)\right)}^{2}} \]

   6. Applied egg-rr 89.3%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 89.3%

         \[\leadsto \ell + \sqrt{\color{blue}{\left(w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right) \cdot \left(w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right)}} \]

      2. rem-sqrt-square 79.1%

         \[\leadsto \ell + \color{blue}{\left|w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right|} \]

      3. sub-neg 79.1%

         \[\leadsto \ell + \left|w \cdot \color{blue}{\left(\log \left({\ell}^{\ell}\right) + \left(-\ell\right)\right)}\right| \]

      4. sub-neg 79.1%

         \[\leadsto \ell + \left|w \cdot \color{blue}{\left(\log \left({\ell}^{\ell}\right) - \ell\right)}\right| \]

      5. log-pow 79.1%

         \[\leadsto \ell + \left|w \cdot \left(\color{blue}{\ell \cdot \log \ell} - \ell\right)\right| \]

   8. Simplified 79.1%

      \[\leadsto \ell + \color{blue}{\left|w \cdot \left(\ell \cdot \log \ell - \ell\right)\right|} \]

   if -4.49999999999999972e262 < w < -8.1000000000000001e34

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 73.7%

      \[\leadsto \frac{\color{blue}{\ell + \ell \cdot \left(w \cdot \log \ell\right)}}{e^{w}} \]
   5. Step-by-step derivation

      1. associate-*r* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in w around inf 73.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}} \]
   8. Step-by-step derivation

      1. associate-*r* 73.7%

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

   9. Simplified 73.7%

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

   if -8.1000000000000001e34 < w < -0.34999999999999998 or 0.0400000000000000008 < w

   1. Initial program 92.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 92.4%

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

      2. associate-*l/ 92.4%

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

      3. *-lft-identity 92.4%

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

   3. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 92.4%

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

      2. pow3 92.3%

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

   5. Applied egg-rr 92.3%

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

   6. Taylor expanded in w around 0 5.6%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 5.6%

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

      2. rem-cube-cbrt 5.6%

         \[\leadsto \color{blue}{\ell} \]

      3. add-log-exp 89.0%

         \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   8. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   if -0.34999999999999998 < w < 0.0400000000000000008

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 99.2%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.5 \cdot 10^{+262}:\\ \;\;\;\;\ell + \left|w \cdot \left(\ell \cdot \log \ell - \ell\right)\right|\\ \mathbf{elif}\;w \leq -8.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -0.35 \lor \neg \left(w \leq 0.04\right):\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \]

Alternative 8: 91.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.55:\\ \;\;\;\;\frac{\ell + \log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \left|w \cdot \left(\ell \cdot \log \ell - \ell\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.55)
   (/ (+ l (* (log l) (* l w))) (exp w))
   (+ l (fabs (* w (- (* l (log l)) l))))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.55) {
		tmp = (l + (log(l) * (l * w))) / exp(w);
	} else {
		tmp = l + fabs((w * ((l * log(l)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 0.55d0) then
        tmp = (l + (log(l) * (l * w))) / exp(w)
    else
        tmp = l + abs((w * ((l * log(l)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (l <= 0.55) {
		tmp = (l + (Math.log(l) * (l * w))) / Math.exp(w);
	} else {
		tmp = l + Math.abs((w * ((l * Math.log(l)) - l)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if l <= 0.55:
		tmp = (l + (math.log(l) * (l * w))) / math.exp(w)
	else:
		tmp = l + math.fabs((w * ((l * math.log(l)) - l)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.55)
		tmp = Float64(Float64(l + Float64(log(l) * Float64(l * w))) / exp(w));
	else
		tmp = Float64(l + abs(Float64(w * Float64(Float64(l * log(l)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (l <= 0.55)
		tmp = (l + (log(l) * (l * w))) / exp(w);
	else
		tmp = l + abs((w * ((l * log(l)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.55], N[(N[(l + N[(N[Log[l], $MachinePrecision] * N[(l * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(l + N[Abs[N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.55000000000000004

   1. Initial program 99.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in w around 0 99.1%

      \[\leadsto \frac{\color{blue}{\ell + \ell \cdot \left(w \cdot \log \ell\right)}}{e^{w}} \]
   5. Step-by-step derivation

      1. associate-*r* 99.1%

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

   6. Simplified 99.1%

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

   if 0.55000000000000004 < l

   1. Initial program 96.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 96.1%

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

      2. associate-*l/ 96.1%

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

      3. *-lft-identity 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in w around 0 75.4%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.1%

         \[\leadsto \ell + \color{blue}{\sqrt{w \cdot \left(\ell \cdot \log \ell - \ell\right)} \cdot \sqrt{w \cdot \left(\ell \cdot \log \ell - \ell\right)}} \]

      2. sqrt-unprod 88.3%

         \[\leadsto \ell + \color{blue}{\sqrt{\left(w \cdot \left(\ell \cdot \log \ell - \ell\right)\right) \cdot \left(w \cdot \left(\ell \cdot \log \ell - \ell\right)\right)}} \]

      3. pow2 88.3%

         \[\leadsto \ell + \sqrt{\color{blue}{{\left(w \cdot \left(\ell \cdot \log \ell - \ell\right)\right)}^{2}}} \]

      4. add-log-exp 29.8%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \left(\color{blue}{\log \left(e^{\ell \cdot \log \ell}\right)} - \ell\right)\right)}^{2}} \]

      5. *-commutative 29.8%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \left(\log \left(e^{\color{blue}{\log \ell \cdot \ell}}\right) - \ell\right)\right)}^{2}} \]

      6. exp-to-pow 29.8%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \left(\log \color{blue}{\left({\ell}^{\ell}\right)} - \ell\right)\right)}^{2}} \]

   6. Applied egg-rr 29.8%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 29.8%

         \[\leadsto \ell + \sqrt{\color{blue}{\left(w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right) \cdot \left(w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right)}} \]

      2. rem-sqrt-square 29.8%

         \[\leadsto \ell + \color{blue}{\left|w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)\right|} \]

      3. sub-neg 29.8%

         \[\leadsto \ell + \left|w \cdot \color{blue}{\left(\log \left({\ell}^{\ell}\right) + \left(-\ell\right)\right)}\right| \]

      4. sub-neg 29.8%

         \[\leadsto \ell + \left|w \cdot \color{blue}{\left(\log \left({\ell}^{\ell}\right) - \ell\right)}\right| \]

      5. log-pow 85.0%

         \[\leadsto \ell + \left|w \cdot \left(\color{blue}{\ell \cdot \log \ell} - \ell\right)\right| \]

   8. Simplified 85.0%

      \[\leadsto \ell + \color{blue}{\left|w \cdot \left(\ell \cdot \log \ell - \ell\right)\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.55:\\ \;\;\;\;\frac{\ell + \log \ell \cdot \left(\ell \cdot w\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \left|w \cdot \left(\ell \cdot \log \ell - \ell\right)\right|\\ \end{array} \]

Alternative 9: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(e^{\ell}\right)\\ \mathbf{if}\;w \leq -3 \cdot 10^{+259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -1.96 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{\ell}}\\ \mathbf{elif}\;w \leq -0.092 \lor \neg \left(w \leq 0.049\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (log (exp l))))
   (if (<= w -3e+259)
     t_0
     (if (<= w -1.96e+64)
       (sqrt (/ (/ 1.0 l) l))
       (if (or (<= w -0.092) (not (<= w 0.049)))
         t_0
         (+ l (* w (- (* l (log l)) l))))))))

C

double code(double w, double l) {
	double t_0 = log(exp(l));
	double tmp;
	if (w <= -3e+259) {
		tmp = t_0;
	} else if (w <= -1.96e+64) {
		tmp = sqrt(((1.0 / l) / l));
	} else if ((w <= -0.092) || !(w <= 0.049)) {
		tmp = t_0;
	} else {
		tmp = l + (w * ((l * log(l)) - l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(exp(l))
    if (w <= (-3d+259)) then
        tmp = t_0
    else if (w <= (-1.96d+64)) then
        tmp = sqrt(((1.0d0 / l) / l))
    else if ((w <= (-0.092d0)) .or. (.not. (w <= 0.049d0))) then
        tmp = t_0
    else
        tmp = l + (w * ((l * log(l)) - l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.log(Math.exp(l));
	double tmp;
	if (w <= -3e+259) {
		tmp = t_0;
	} else if (w <= -1.96e+64) {
		tmp = Math.sqrt(((1.0 / l) / l));
	} else if ((w <= -0.092) || !(w <= 0.049)) {
		tmp = t_0;
	} else {
		tmp = l + (w * ((l * Math.log(l)) - l));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.log(math.exp(l))
	tmp = 0
	if w <= -3e+259:
		tmp = t_0
	elif w <= -1.96e+64:
		tmp = math.sqrt(((1.0 / l) / l))
	elif (w <= -0.092) or not (w <= 0.049):
		tmp = t_0
	else:
		tmp = l + (w * ((l * math.log(l)) - l))
	return tmp

Julia

function code(w, l)
	t_0 = log(exp(l))
	tmp = 0.0
	if (w <= -3e+259)
		tmp = t_0;
	elseif (w <= -1.96e+64)
		tmp = sqrt(Float64(Float64(1.0 / l) / l));
	elseif ((w <= -0.092) || !(w <= 0.049))
		tmp = t_0;
	else
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = log(exp(l));
	tmp = 0.0;
	if (w <= -3e+259)
		tmp = t_0;
	elseif (w <= -1.96e+64)
		tmp = sqrt(((1.0 / l) / l));
	elseif ((w <= -0.092) || ~((w <= 0.049)))
		tmp = t_0;
	else
		tmp = l + (w * ((l * log(l)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Log[N[Exp[l], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[w, -3e+259], t$95$0, If[LessEqual[w, -1.96e+64], N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[w, -0.092], N[Not[LessEqual[w, 0.049]], $MachinePrecision]], t$95$0, N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -3.00000000000000013e259 or -1.9599999999999999e64 < w < -0.091999999999999998 or 0.049000000000000002 < w

   1. Initial program 94.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 94.2%

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

      2. associate-*l/ 94.2%

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

      3. *-lft-identity 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 94.2%

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

      2. pow3 94.2%

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

   5. Applied egg-rr 94.2%

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

   6. Taylor expanded in w around 0 5.1%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 5.1%

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

      2. rem-cube-cbrt 5.1%

         \[\leadsto \color{blue}{\ell} \]

      3. add-log-exp 80.4%

         \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   8. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\log \left(e^{\ell}\right)} \]

   if -3.00000000000000013e259 < w < -1.9599999999999999e64

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around 0 2.9%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 2.9%

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

      2. rem-cube-cbrt 2.9%

         \[\leadsto \color{blue}{\ell} \]

      3. add-exp-log 2.9%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 5.5%

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

      6. sqr-neg 5.5%

         \[\leadsto e^{\sqrt{\color{blue}{\left(-\log \ell\right) \cdot \left(-\log \ell\right)}}} \]

      7. sqrt-unprod 4.3%

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

      8. add-sqr-sqrt 4.9%

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

      9. rec-exp 4.9%

         \[\leadsto \color{blue}{\frac{1}{e^{\log \ell}}} \]

      10. add-exp-log 4.9%

          \[\leadsto \frac{1}{\color{blue}{\ell}} \]

      11. add-sqr-sqrt 4.9%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      12. associate-/r* 4.9%

          \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]

   8. Applied egg-rr 4.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 4.9%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}} \]

      2. sqrt-unprod 41.9%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}} \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}} \]

      3. frac-times 41.9%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\sqrt{\ell}} \cdot \frac{1}{\sqrt{\ell}}}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \]

      4. div-inv 41.9%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. associate-/l/ 41.9%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{\sqrt{\ell} \cdot \sqrt{\ell}}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      6. add-sqr-sqrt 41.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\color{blue}{\ell}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      7. add-sqr-sqrt 41.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{\color{blue}{\ell}}} \]

   10. Applied egg-rr 41.9%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{\ell}}} \]

   if -0.091999999999999998 < w < 0.049000000000000002

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 99.2%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3 \cdot 10^{+259}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{elif}\;w \leq -1.96 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{\ell}}\\ \mathbf{elif}\;w \leq -0.092 \lor \neg \left(w \leq 0.049\right):\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \]

Alternative 10: 70.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.62:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{\ell}}\\ \mathbf{elif}\;w \leq 0.0125:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.62)
   (sqrt (/ (/ 1.0 l) l))
   (if (<= w 0.0125) (+ l (* w (- (* l (log l)) l))) (sqrt (* l l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.62) {
		tmp = sqrt(((1.0 / l) / l));
	} else if (w <= 0.0125) {
		tmp = l + (w * ((l * log(l)) - l));
	} else {
		tmp = sqrt((l * l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.62d0)) then
        tmp = sqrt(((1.0d0 / l) / l))
    else if (w <= 0.0125d0) then
        tmp = l + (w * ((l * log(l)) - l))
    else
        tmp = sqrt((l * l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.62) {
		tmp = Math.sqrt(((1.0 / l) / l));
	} else if (w <= 0.0125) {
		tmp = l + (w * ((l * Math.log(l)) - l));
	} else {
		tmp = Math.sqrt((l * l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.62:
		tmp = math.sqrt(((1.0 / l) / l))
	elif w <= 0.0125:
		tmp = l + (w * ((l * math.log(l)) - l))
	else:
		tmp = math.sqrt((l * l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.62)
		tmp = sqrt(Float64(Float64(1.0 / l) / l));
	elseif (w <= 0.0125)
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	else
		tmp = sqrt(Float64(l * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.62)
		tmp = sqrt(((1.0 / l) / l));
	elseif (w <= 0.0125)
		tmp = l + (w * ((l * log(l)) - l));
	else
		tmp = sqrt((l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.62], N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.0125], N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(l * l), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -0.619999999999999996

   1. Initial program 99.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 3.6%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 3.6%

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

      2. rem-cube-cbrt 3.6%

         \[\leadsto \color{blue}{\ell} \]

      3. add-exp-log 3.6%

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

      4. add-sqr-sqrt 2.3%

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

      5. sqrt-unprod 5.3%

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

      6. sqr-neg 5.3%

         \[\leadsto e^{\sqrt{\color{blue}{\left(-\log \ell\right) \cdot \left(-\log \ell\right)}}} \]

      7. sqrt-unprod 3.0%

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

      8. add-sqr-sqrt 4.2%

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

      9. rec-exp 4.2%

         \[\leadsto \color{blue}{\frac{1}{e^{\log \ell}}} \]

      10. add-exp-log 4.2%

          \[\leadsto \frac{1}{\color{blue}{\ell}} \]

      11. add-sqr-sqrt 4.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      12. associate-/r* 4.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]

   8. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 4.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}} \]

      2. sqrt-unprod 30.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}} \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}} \]

      3. frac-times 30.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\sqrt{\ell}} \cdot \frac{1}{\sqrt{\ell}}}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \]

      4. div-inv 30.2%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. associate-/l/ 30.2%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{\sqrt{\ell} \cdot \sqrt{\ell}}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      6. add-sqr-sqrt 30.2%

         \[\leadsto \sqrt{\frac{\frac{1}{\color{blue}{\ell}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      7. add-sqr-sqrt 30.2%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{\color{blue}{\ell}}} \]

   10. Applied egg-rr 30.2%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{\ell}}} \]

   if -0.619999999999999996 < w < 0.012500000000000001

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 99.2%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

   if 0.012500000000000001 < w

   1. Initial program 90.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 90.7%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 90.7%

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

      2. pow3 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in w around 0 5.6%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 5.6%

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

      2. rem-cube-cbrt 5.6%

         \[\leadsto \color{blue}{\ell} \]

      3. add-sqr-sqrt 5.6%

         \[\leadsto \color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}} \]

      4. sqrt-unprod 55.2%

         \[\leadsto \color{blue}{\sqrt{\ell \cdot \ell}} \]

   8. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{\sqrt{\ell \cdot \ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.62:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{\ell}}\\ \mathbf{elif}\;w \leq 0.0125:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \ell}\\ \end{array} \]

Alternative 11: 69.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.8 \cdot 10^{+25} \lor \neg \left(w \leq 0.058\right):\\ \;\;\;\;\sqrt{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -3.8e+25) (not (<= w 0.058))) (sqrt (* l l)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -3.8e+25) || !(w <= 0.058)) {
		tmp = sqrt((l * l));
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-3.8d+25)) .or. (.not. (w <= 0.058d0))) then
        tmp = sqrt((l * l))
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -3.8e+25) || !(w <= 0.058)) {
		tmp = Math.sqrt((l * l));
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -3.8e+25) or not (w <= 0.058):
		tmp = math.sqrt((l * l))
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -3.8e+25) || !(w <= 0.058))
		tmp = sqrt(Float64(l * l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -3.8e+25) || ~((w <= 0.058)))
		tmp = sqrt((l * l));
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -3.8e+25], N[Not[LessEqual[w, 0.058]], $MachinePrecision]], N[Sqrt[N[(l * l), $MachinePrecision]], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -3.8e25 or 0.0580000000000000029 < w

   1. Initial program 95.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 95.7%

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

      2. associate-*l/ 95.7%

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

      3. *-lft-identity 95.7%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 95.7%

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

      2. pow3 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in w around 0 4.4%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 4.4%

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

      2. rem-cube-cbrt 4.4%

         \[\leadsto \color{blue}{\ell} \]

      3. add-sqr-sqrt 4.4%

         \[\leadsto \color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}} \]

      4. sqrt-unprod 36.0%

         \[\leadsto \color{blue}{\sqrt{\ell \cdot \ell}} \]

   8. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{\sqrt{\ell \cdot \ell}} \]

   if -3.8e25 < w < 0.0580000000000000029

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 95.0%

      \[\leadsto \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.8 \cdot 10^{+25} \lor \neg \left(w \leq 0.058\right):\\ \;\;\;\;\sqrt{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 12: 70.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -62:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{\ell}}\\ \mathbf{elif}\;w \leq 0.075:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -62.0) (sqrt (/ (/ 1.0 l) l)) (if (<= w 0.075) l (sqrt (* l l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -62.0) {
		tmp = sqrt(((1.0 / l) / l));
	} else if (w <= 0.075) {
		tmp = l;
	} else {
		tmp = sqrt((l * l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-62.0d0)) then
        tmp = sqrt(((1.0d0 / l) / l))
    else if (w <= 0.075d0) then
        tmp = l
    else
        tmp = sqrt((l * l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -62.0) {
		tmp = Math.sqrt(((1.0 / l) / l));
	} else if (w <= 0.075) {
		tmp = l;
	} else {
		tmp = Math.sqrt((l * l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -62.0:
		tmp = math.sqrt(((1.0 / l) / l))
	elif w <= 0.075:
		tmp = l
	else:
		tmp = math.sqrt((l * l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -62.0)
		tmp = sqrt(Float64(Float64(1.0 / l) / l));
	elseif (w <= 0.075)
		tmp = l;
	else
		tmp = sqrt(Float64(l * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -62.0)
		tmp = sqrt(((1.0 / l) / l));
	elseif (w <= 0.075)
		tmp = l;
	else
		tmp = sqrt((l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -62.0], N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.075], l, N[Sqrt[N[(l * l), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -62

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around 0 3.4%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 3.4%

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

      2. rem-cube-cbrt 3.4%

         \[\leadsto \color{blue}{\ell} \]

      3. add-exp-log 3.4%

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

      4. add-sqr-sqrt 2.1%

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

      5. sqrt-unprod 5.2%

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

      6. sqr-neg 5.2%

         \[\leadsto e^{\sqrt{\color{blue}{\left(-\log \ell\right) \cdot \left(-\log \ell\right)}}} \]

      7. sqrt-unprod 3.2%

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

      8. add-sqr-sqrt 4.1%

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

      9. rec-exp 4.1%

         \[\leadsto \color{blue}{\frac{1}{e^{\log \ell}}} \]

      10. add-exp-log 4.1%

          \[\leadsto \frac{1}{\color{blue}{\ell}} \]

      11. add-sqr-sqrt 4.1%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      12. associate-/r* 4.1%

          \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]

   8. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 4.1%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}} \]

      2. sqrt-unprod 31.1%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}} \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}} \]

      3. frac-times 31.1%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\sqrt{\ell}} \cdot \frac{1}{\sqrt{\ell}}}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \]

      4. div-inv 31.1%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      5. associate-/l/ 31.1%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{\sqrt{\ell} \cdot \sqrt{\ell}}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      6. add-sqr-sqrt 31.1%

         \[\leadsto \sqrt{\frac{\frac{1}{\color{blue}{\ell}}}{\sqrt{\ell} \cdot \sqrt{\ell}}} \]

      7. add-sqr-sqrt 31.1%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{\color{blue}{\ell}}} \]

   10. Applied egg-rr 31.1%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{\ell}}} \]

   if -62 < w < 0.0749999999999999972

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 97.4%

      \[\leadsto \color{blue}{\ell} \]

   if 0.0749999999999999972 < w

   1. Initial program 90.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 90.7%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 90.7%

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

      2. pow3 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in w around 0 5.6%

      \[\leadsto {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 5.6%

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

      2. rem-cube-cbrt 5.6%

         \[\leadsto \color{blue}{\ell} \]

      3. add-sqr-sqrt 5.6%

         \[\leadsto \color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}} \]

      4. sqrt-unprod 55.2%

         \[\leadsto \color{blue}{\sqrt{\ell \cdot \ell}} \]

   8. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{\sqrt{\ell \cdot \ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -62:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{\ell}}\\ \mathbf{elif}\;w \leq 0.075:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell \cdot \ell}\\ \end{array} \]

Alternative 13: 56.8% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ \ell \end{array} \]

FPCore

(FPCore (w l) :precision binary64 l)

C

double code(double w, double l) {
	return l;
}

Fortran

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

Java

public static double code(double w, double l) {
	return l;
}

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

function tmp = code(w, l)
	tmp = l;
end

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 98.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in w around 0 61.8%

   \[\leadsto \color{blue}{\ell} \]

5. Final simplification 61.8%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023279 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))