exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 31.2s

Alternatives: 7

Speedup: N/A×

• 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: 99.4% 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: 99.4% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]
6. Add Preprocessing

Alternative 2: 84.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ t_1 := \frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{if}\;w \leq -1.25 \cdot 10^{+236}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq -5 \cdot 10^{+195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -2 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq -3.7 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -11.2 \lor \neg \left(w \leq 0.007\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (/ (* (* l w) (log l)) (exp w)))
        (t_1 (/ (* (log l) (* w (- l))) (exp w))))
   (if (<= w -1.25e+236)
     t_1
     (if (<= w -5e+195)
       t_0
       (if (<= w -2e+141)
         t_1
         (if (<= w -3.7e+112)
           t_0
           (if (or (<= w -11.2) (not (<= w 0.007)))
             t_1
             (+ l (* l (* w (+ (log l) -1.0)))))))))))

C

double code(double w, double l) {
	double t_0 = ((l * w) * log(l)) / exp(w);
	double t_1 = (log(l) * (w * -l)) / exp(w);
	double tmp;
	if (w <= -1.25e+236) {
		tmp = t_1;
	} else if (w <= -5e+195) {
		tmp = t_0;
	} else if (w <= -2e+141) {
		tmp = t_1;
	} else if (w <= -3.7e+112) {
		tmp = t_0;
	} else if ((w <= -11.2) || !(w <= 0.007)) {
		tmp = t_1;
	} else {
		tmp = l + (l * (w * (log(l) + -1.0)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((l * w) * log(l)) / exp(w)
    t_1 = (log(l) * (w * -l)) / exp(w)
    if (w <= (-1.25d+236)) then
        tmp = t_1
    else if (w <= (-5d+195)) then
        tmp = t_0
    else if (w <= (-2d+141)) then
        tmp = t_1
    else if (w <= (-3.7d+112)) then
        tmp = t_0
    else if ((w <= (-11.2d0)) .or. (.not. (w <= 0.007d0))) then
        tmp = t_1
    else
        tmp = l + (l * (w * (log(l) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = ((l * w) * Math.log(l)) / Math.exp(w);
	double t_1 = (Math.log(l) * (w * -l)) / Math.exp(w);
	double tmp;
	if (w <= -1.25e+236) {
		tmp = t_1;
	} else if (w <= -5e+195) {
		tmp = t_0;
	} else if (w <= -2e+141) {
		tmp = t_1;
	} else if (w <= -3.7e+112) {
		tmp = t_0;
	} else if ((w <= -11.2) || !(w <= 0.007)) {
		tmp = t_1;
	} else {
		tmp = l + (l * (w * (Math.log(l) + -1.0)));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = ((l * w) * math.log(l)) / math.exp(w)
	t_1 = (math.log(l) * (w * -l)) / math.exp(w)
	tmp = 0
	if w <= -1.25e+236:
		tmp = t_1
	elif w <= -5e+195:
		tmp = t_0
	elif w <= -2e+141:
		tmp = t_1
	elif w <= -3.7e+112:
		tmp = t_0
	elif (w <= -11.2) or not (w <= 0.007):
		tmp = t_1
	else:
		tmp = l + (l * (w * (math.log(l) + -1.0)))
	return tmp

Julia

function code(w, l)
	t_0 = Float64(Float64(Float64(l * w) * log(l)) / exp(w))
	t_1 = Float64(Float64(log(l) * Float64(w * Float64(-l))) / exp(w))
	tmp = 0.0
	if (w <= -1.25e+236)
		tmp = t_1;
	elseif (w <= -5e+195)
		tmp = t_0;
	elseif (w <= -2e+141)
		tmp = t_1;
	elseif (w <= -3.7e+112)
		tmp = t_0;
	elseif ((w <= -11.2) || !(w <= 0.007))
		tmp = t_1;
	else
		tmp = Float64(l + Float64(l * Float64(w * Float64(log(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = ((l * w) * log(l)) / exp(w);
	t_1 = (log(l) * (w * -l)) / exp(w);
	tmp = 0.0;
	if (w <= -1.25e+236)
		tmp = t_1;
	elseif (w <= -5e+195)
		tmp = t_0;
	elseif (w <= -2e+141)
		tmp = t_1;
	elseif (w <= -3.7e+112)
		tmp = t_0;
	elseif ((w <= -11.2) || ~((w <= 0.007)))
		tmp = t_1;
	else
		tmp = l + (l * (w * (log(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(N[(l * w), $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[l], $MachinePrecision] * N[(w * (-l)), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.25e+236], t$95$1, If[LessEqual[w, -5e+195], t$95$0, If[LessEqual[w, -2e+141], t$95$1, If[LessEqual[w, -3.7e+112], t$95$0, If[Or[LessEqual[w, -11.2], N[Not[LessEqual[w, 0.007]], $MachinePrecision]], t$95$1, N[(l + N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -1.24999999999999993e236 or -4.9999999999999998e195 < w < -2.00000000000000003e141 or -3.70000000000000004e112 < w < -11.199999999999999 or 0.00700000000000000015 < w

   1. Initial program 98.7%

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

      1. exp-neg 98.7%

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

      2. remove-double-neg 98.7%

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

      3. associate-*l/ 98.7%

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

      4. *-lft-identity 98.7%

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

      5. remove-double-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in w around 0 46.2%

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

      1. associate-*r* 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in w around inf 46.2%

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

      1. associate-*r* 46.2%

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

      2. *-commutative 46.2%

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

   10. Simplified 46.2%

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

       1. add-sqr-sqrt 46.2%

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

       2. sqrt-unprod 36.1%

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

       3. sqr-neg 36.1%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 76.6%

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

       6. distribute-lft-neg-out 76.6%

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

       7. distribute-rgt-neg-in 76.6%

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

       8. neg-sub0 76.6%

          \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]

   12. Applied egg-rr 76.6%

       \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]
   13. Step-by-step derivation

       1. neg-sub0 76.6%

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

       2. distribute-rgt-neg-in 76.6%

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

   14. Simplified 76.6%

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

   if -1.24999999999999993e236 < w < -4.9999999999999998e195 or -2.00000000000000003e141 < w < -3.70000000000000004e112

   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. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 77.8%

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

      1. associate-*r* 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in w around inf 77.8%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

   10. Simplified 77.8%

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

   if -11.199999999999999 < w < 0.00700000000000000015

   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. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \ell + \color{blue}{\ell \cdot \left(w \cdot \left(\log \ell - 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.25 \cdot 10^{+236}:\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{elif}\;w \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -3.7 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{elif}\;w \leq -11.2 \lor \neg \left(w \leq 0.007\right):\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ t_1 := \frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ t_2 := \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \mathbf{if}\;w \leq -1.3 \cdot 10^{+236}:\\ \;\;\;\;\ell + \left|t\_2\right|\\ \mathbf{elif}\;w \leq -4 \cdot 10^{+196}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -2 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq -1 \cdot 10^{+111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -0.25 \lor \neg \left(w \leq 0.007\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\ell + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (/ (* (* l w) (log l)) (exp w)))
        (t_1 (/ (* (log l) (* w (- l))) (exp w)))
        (t_2 (* l (* w (+ (log l) -1.0)))))
   (if (<= w -1.3e+236)
     (+ l (fabs t_2))
     (if (<= w -4e+196)
       t_0
       (if (<= w -2e+141)
         t_1
         (if (<= w -1e+111)
           t_0
           (if (or (<= w -0.25) (not (<= w 0.007))) t_1 (+ l t_2))))))))

C

double code(double w, double l) {
	double t_0 = ((l * w) * log(l)) / exp(w);
	double t_1 = (log(l) * (w * -l)) / exp(w);
	double t_2 = l * (w * (log(l) + -1.0));
	double tmp;
	if (w <= -1.3e+236) {
		tmp = l + fabs(t_2);
	} else if (w <= -4e+196) {
		tmp = t_0;
	} else if (w <= -2e+141) {
		tmp = t_1;
	} else if (w <= -1e+111) {
		tmp = t_0;
	} else if ((w <= -0.25) || !(w <= 0.007)) {
		tmp = t_1;
	} else {
		tmp = l + t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((l * w) * log(l)) / exp(w)
    t_1 = (log(l) * (w * -l)) / exp(w)
    t_2 = l * (w * (log(l) + (-1.0d0)))
    if (w <= (-1.3d+236)) then
        tmp = l + abs(t_2)
    else if (w <= (-4d+196)) then
        tmp = t_0
    else if (w <= (-2d+141)) then
        tmp = t_1
    else if (w <= (-1d+111)) then
        tmp = t_0
    else if ((w <= (-0.25d0)) .or. (.not. (w <= 0.007d0))) then
        tmp = t_1
    else
        tmp = l + t_2
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = ((l * w) * Math.log(l)) / Math.exp(w);
	double t_1 = (Math.log(l) * (w * -l)) / Math.exp(w);
	double t_2 = l * (w * (Math.log(l) + -1.0));
	double tmp;
	if (w <= -1.3e+236) {
		tmp = l + Math.abs(t_2);
	} else if (w <= -4e+196) {
		tmp = t_0;
	} else if (w <= -2e+141) {
		tmp = t_1;
	} else if (w <= -1e+111) {
		tmp = t_0;
	} else if ((w <= -0.25) || !(w <= 0.007)) {
		tmp = t_1;
	} else {
		tmp = l + t_2;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = ((l * w) * math.log(l)) / math.exp(w)
	t_1 = (math.log(l) * (w * -l)) / math.exp(w)
	t_2 = l * (w * (math.log(l) + -1.0))
	tmp = 0
	if w <= -1.3e+236:
		tmp = l + math.fabs(t_2)
	elif w <= -4e+196:
		tmp = t_0
	elif w <= -2e+141:
		tmp = t_1
	elif w <= -1e+111:
		tmp = t_0
	elif (w <= -0.25) or not (w <= 0.007):
		tmp = t_1
	else:
		tmp = l + t_2
	return tmp

Julia

function code(w, l)
	t_0 = Float64(Float64(Float64(l * w) * log(l)) / exp(w))
	t_1 = Float64(Float64(log(l) * Float64(w * Float64(-l))) / exp(w))
	t_2 = Float64(l * Float64(w * Float64(log(l) + -1.0)))
	tmp = 0.0
	if (w <= -1.3e+236)
		tmp = Float64(l + abs(t_2));
	elseif (w <= -4e+196)
		tmp = t_0;
	elseif (w <= -2e+141)
		tmp = t_1;
	elseif (w <= -1e+111)
		tmp = t_0;
	elseif ((w <= -0.25) || !(w <= 0.007))
		tmp = t_1;
	else
		tmp = Float64(l + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = ((l * w) * log(l)) / exp(w);
	t_1 = (log(l) * (w * -l)) / exp(w);
	t_2 = l * (w * (log(l) + -1.0));
	tmp = 0.0;
	if (w <= -1.3e+236)
		tmp = l + abs(t_2);
	elseif (w <= -4e+196)
		tmp = t_0;
	elseif (w <= -2e+141)
		tmp = t_1;
	elseif (w <= -1e+111)
		tmp = t_0;
	elseif ((w <= -0.25) || ~((w <= 0.007)))
		tmp = t_1;
	else
		tmp = l + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(N[(l * w), $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[l], $MachinePrecision] * N[(w * (-l)), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.3e+236], N[(l + N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -4e+196], t$95$0, If[LessEqual[w, -2e+141], t$95$1, If[LessEqual[w, -1e+111], t$95$0, If[Or[LessEqual[w, -0.25], N[Not[LessEqual[w, 0.007]], $MachinePrecision]], t$95$1, N[(l + t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if w < -1.3e236

   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. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 79.9%

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

      3. pow2 79.9%

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

      4. sub-neg 79.9%

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

      5. distribute-rgt-in 21.5%

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

      6. neg-mul-1 21.5%

         \[\leadsto \ell + \sqrt{{\left(\left(\ell \cdot \log \ell\right) \cdot w + \color{blue}{\left(-1 \cdot \ell\right)} \cdot w\right)}^{2}} \]

      7. *-commutative 21.5%

         \[\leadsto \ell + \sqrt{{\left(\left(\ell \cdot \log \ell\right) \cdot w + \color{blue}{\left(\ell \cdot -1\right)} \cdot w\right)}^{2}} \]

      8. distribute-rgt-in 79.9%

         \[\leadsto \ell + \sqrt{{\color{blue}{\left(w \cdot \left(\ell \cdot \log \ell + \ell \cdot -1\right)\right)}}^{2}} \]

      9. distribute-lft-in 79.9%

         \[\leadsto \ell + \sqrt{{\left(w \cdot \color{blue}{\left(\ell \cdot \left(\log \ell + -1\right)\right)}\right)}^{2}} \]

      10. associate-*r* 79.9%

          \[\leadsto \ell + \sqrt{{\color{blue}{\left(\left(w \cdot \ell\right) \cdot \left(\log \ell + -1\right)\right)}}^{2}} \]

      11. *-commutative 79.9%

          \[\leadsto \ell + \sqrt{{\left(\color{blue}{\left(\ell \cdot w\right)} \cdot \left(\log \ell + -1\right)\right)}^{2}} \]

   7. Applied egg-rr 79.9%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(\left(\ell \cdot w\right) \cdot \left(\log \ell + -1\right)\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 79.9%

         \[\leadsto \ell + \sqrt{\color{blue}{\left(\left(\ell \cdot w\right) \cdot \left(\log \ell + -1\right)\right) \cdot \left(\left(\ell \cdot w\right) \cdot \left(\log \ell + -1\right)\right)}} \]

      2. rem-sqrt-square 60.6%

         \[\leadsto \ell + \color{blue}{\left|\left(\ell \cdot w\right) \cdot \left(\log \ell + -1\right)\right|} \]

      3. associate-*r* 64.6%

         \[\leadsto \ell + \left|\color{blue}{\ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)}\right| \]

      4. +-commutative 64.6%

         \[\leadsto \ell + \left|\ell \cdot \left(w \cdot \color{blue}{\left(-1 + \log \ell\right)}\right)\right| \]

   9. Simplified 64.6%

      \[\leadsto \ell + \color{blue}{\left|\ell \cdot \left(w \cdot \left(-1 + \log \ell\right)\right)\right|} \]

   if -1.3e236 < w < -3.9999999999999998e196 or -2.00000000000000003e141 < w < -9.99999999999999957e110

   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. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 77.8%

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

      1. associate-*r* 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in w around inf 77.8%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

   10. Simplified 77.8%

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

   if -3.9999999999999998e196 < w < -2.00000000000000003e141 or -9.99999999999999957e110 < w < -0.25 or 0.00700000000000000015 < w

   1. Initial program 98.3%

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

      1. exp-neg 98.3%

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

      2. remove-double-neg 98.3%

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

      3. associate-*l/ 98.3%

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

      4. *-lft-identity 98.3%

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

      5. remove-double-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in w around 0 49.4%

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

      1. associate-*r* 49.4%

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

   7. Simplified 49.4%

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

   8. Taylor expanded in w around inf 49.4%

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

      1. associate-*r* 49.4%

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

      2. *-commutative 49.4%

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

   10. Simplified 49.4%

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

       1. add-sqr-sqrt 49.4%

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

       2. sqrt-unprod 43.3%

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

       3. sqr-neg 43.3%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 81.8%

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

       6. distribute-lft-neg-out 81.8%

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

       7. distribute-rgt-neg-in 81.8%

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

       8. neg-sub0 81.8%

          \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]

   12. Applied egg-rr 81.8%

       \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]
   13. Step-by-step derivation

       1. neg-sub0 81.8%

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

       2. distribute-rgt-neg-in 81.8%

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

   14. Simplified 81.8%

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

   if -0.25 < w < 0.00700000000000000015

   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. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \ell + \color{blue}{\ell \cdot \left(w \cdot \left(\log \ell - 1\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.3 \cdot 10^{+236}:\\ \;\;\;\;\ell + \left|\ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\right|\\ \mathbf{elif}\;w \leq -4 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{elif}\;w \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{elif}\;w \leq -1 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{elif}\;w \leq -0.25 \lor \neg \left(w \leq 0.007\right):\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.05 \cdot 10^{+238} \lor \neg \left(w \leq -1.55 \cdot 10^{+109}\right) \land w \leq 1:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -1.05e+238) (and (not (<= w -1.55e+109)) (<= w 1.0)))
   (* l (- 1.0 w))
   (/ (* (* l w) (log l)) (exp w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -1.05e+238) || (!(w <= -1.55e+109) && (w <= 1.0))) {
		tmp = l * (1.0 - w);
	} else {
		tmp = ((l * w) * log(l)) / 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 <= (-1.05d+238)) .or. (.not. (w <= (-1.55d+109))) .and. (w <= 1.0d0)) then
        tmp = l * (1.0d0 - w)
    else
        tmp = ((l * w) * log(l)) / exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -1.05e+238) || (!(w <= -1.55e+109) && (w <= 1.0))) {
		tmp = l * (1.0 - w);
	} else {
		tmp = ((l * w) * Math.log(l)) / Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -1.05e+238) or (not (w <= -1.55e+109) and (w <= 1.0)):
		tmp = l * (1.0 - w)
	else:
		tmp = ((l * w) * math.log(l)) / math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -1.05e+238) || (!(w <= -1.55e+109) && (w <= 1.0)))
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(Float64(Float64(l * w) * log(l)) / exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -1.05e+238) || (~((w <= -1.55e+109)) && (w <= 1.0)))
		tmp = l * (1.0 - w);
	else
		tmp = ((l * w) * log(l)) / exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -1.05e+238], And[N[Not[LessEqual[w, -1.55e+109]], $MachinePrecision], LessEqual[w, 1.0]]], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * w), $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.05000000000000004e238 or -1.54999999999999996e109 < w < 1

   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. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 74.1%

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

      1. add-log-exp 39.5%

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

      2. *-commutative 39.5%

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

      3. exp-to-pow 39.5%

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

   7. Applied egg-rr 39.5%

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

   8. Taylor expanded in l around inf 84.1%

      \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
   9. Step-by-step derivation

      1. neg-mul-1 84.1%

         \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

      2. unsub-neg 84.1%

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

   10. Simplified 84.1%

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

   if -1.05000000000000004e238 < w < -1.54999999999999996e109 or 1 < w

   1. Initial program 98.1%

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

      1. exp-neg 98.1%

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

      2. remove-double-neg 98.1%

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

      3. associate-*l/ 98.1%

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

      4. *-lft-identity 98.1%

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

      5. remove-double-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in w around 0 74.2%

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

      1. associate-*r* 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in w around inf 74.2%

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

      1. associate-*r* 74.2%

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

      2. *-commutative 74.2%

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

   10. Simplified 74.2%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.05 \cdot 10^{+238} \lor \neg \left(w \leq -1.55 \cdot 10^{+109}\right) \land w \leq 1:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell + \left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{elif}\;\ell \leq 10^{+39}:\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 - \log \ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.8e+21)
   (/ (+ l (* (* l w) (log l))) (exp w))
   (if (<= l 1e+39)
     (/ (* (log l) (* w (- l))) (exp w))
     (* l (+ 1.0 (* w (- -1.0 (log l))))))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 1.8e+21) {
		tmp = (l + ((l * w) * log(l))) / exp(w);
	} else if (l <= 1e+39) {
		tmp = (log(l) * (w * -l)) / exp(w);
	} else {
		tmp = l * (1.0 + (w * (-1.0 - log(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 <= 1.8d+21) then
        tmp = (l + ((l * w) * log(l))) / exp(w)
    else if (l <= 1d+39) then
        tmp = (log(l) * (w * -l)) / exp(w)
    else
        tmp = l * (1.0d0 + (w * ((-1.0d0) - log(l))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (l <= 1.8e+21) {
		tmp = (l + ((l * w) * Math.log(l))) / Math.exp(w);
	} else if (l <= 1e+39) {
		tmp = (Math.log(l) * (w * -l)) / Math.exp(w);
	} else {
		tmp = l * (1.0 + (w * (-1.0 - Math.log(l))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if l <= 1.8e+21:
		tmp = (l + ((l * w) * math.log(l))) / math.exp(w)
	elif l <= 1e+39:
		tmp = (math.log(l) * (w * -l)) / math.exp(w)
	else:
		tmp = l * (1.0 + (w * (-1.0 - math.log(l))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 1.8e+21)
		tmp = Float64(Float64(l + Float64(Float64(l * w) * log(l))) / exp(w));
	elseif (l <= 1e+39)
		tmp = Float64(Float64(log(l) * Float64(w * Float64(-l))) / exp(w));
	else
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 - log(l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (l <= 1.8e+21)
		tmp = (l + ((l * w) * log(l))) / exp(w);
	elseif (l <= 1e+39)
		tmp = (log(l) * (w * -l)) / exp(w);
	else
		tmp = l * (1.0 + (w * (-1.0 - log(l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[l, 1.8e+21], N[(N[(l + N[(N[(l * w), $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+39], N[(N[(N[Log[l], $MachinePrecision] * N[(w * (-l)), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(l * N[(1.0 + N[(w * N[(-1.0 - N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.8e21

   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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 97.3%

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

      1. associate-*r* 97.3%

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

   7. Simplified 97.3%

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

   if 1.8e21 < l < 9.9999999999999994e38

   1. Initial program 90.0%

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

      1. exp-neg 90.0%

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

      2. remove-double-neg 90.0%

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

      3. associate-*l/ 90.0%

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

      4. *-lft-identity 90.0%

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

      5. remove-double-neg 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in w around 0 20.2%

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

      1. associate-*r* 20.2%

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

   7. Simplified 20.2%

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

   8. Taylor expanded in w around inf 0.6%

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

      1. associate-*r* 0.6%

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

      2. *-commutative 0.6%

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

   10. Simplified 0.6%

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

       1. add-sqr-sqrt 0.6%

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

       2. sqrt-unprod 0.6%

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

       3. sqr-neg 0.6%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 71.0%

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

       6. distribute-lft-neg-out 71.0%

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

       7. distribute-rgt-neg-in 71.0%

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

       8. neg-sub0 71.0%

          \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]

   12. Applied egg-rr 71.0%

       \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]
   13. Step-by-step derivation

       1. neg-sub0 71.0%

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

       2. distribute-rgt-neg-in 71.0%

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

   14. Simplified 71.0%

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

   if 9.9999999999999994e38 < l

   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. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 60.0%

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

      1. sub-neg 60.0%

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

      2. distribute-rgt-in 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-*r* 60.0%

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

   7. Applied egg-rr 60.0%

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

      1. add-sqr-sqrt 1.9%

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

      2. sqrt-unprod 2.0%

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

      3. sqr-neg 2.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 41.9%

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

      6. distribute-lft-neg-out 41.9%

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

      7. distribute-rgt-neg-in 41.9%

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

      8. neg-sub0 41.9%

         \[\leadsto \frac{\color{blue}{0 - \log \ell \cdot \left(\ell \cdot w\right)}}{e^{w}} \]

   9. Applied egg-rr 85.0%

      \[\leadsto \ell + \left(\color{blue}{\left(0 - \log \ell \cdot \left(\ell \cdot w\right)\right)} + \left(-\ell\right) \cdot w\right) \]
   10. Step-by-step derivation

       1. neg-sub0 41.9%

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

       2. distribute-rgt-neg-in 41.9%

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

   11. Simplified 85.0%

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

   12. Taylor expanded in l around 0 85.0%

       \[\leadsto \color{blue}{\ell \cdot \left(1 + \left(-1 \cdot w + -1 \cdot \left(w \cdot \log \ell\right)\right)\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-in 85.0%

          \[\leadsto \color{blue}{\ell \cdot 1 + \ell \cdot \left(-1 \cdot w + -1 \cdot \left(w \cdot \log \ell\right)\right)} \]

       2. *-rgt-identity 85.0%

          \[\leadsto \color{blue}{\ell} + \ell \cdot \left(-1 \cdot w + -1 \cdot \left(w \cdot \log \ell\right)\right) \]

       3. neg-mul-1 85.0%

          \[\leadsto \ell + \ell \cdot \left(\color{blue}{\left(-w\right)} + -1 \cdot \left(w \cdot \log \ell\right)\right) \]

       4. +-commutative 85.0%

          \[\leadsto \ell + \ell \cdot \color{blue}{\left(-1 \cdot \left(w \cdot \log \ell\right) + \left(-w\right)\right)} \]

       5. distribute-lft-in 85.0%

          \[\leadsto \ell + \color{blue}{\left(\ell \cdot \left(-1 \cdot \left(w \cdot \log \ell\right)\right) + \ell \cdot \left(-w\right)\right)} \]

       6. mul-1-neg 85.0%

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

       7. distribute-rgt-neg-in 85.0%

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

       8. associate-*r* 85.0%

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

       9. *-commutative 85.0%

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

       10. associate-*r* 85.0%

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

       11. *-commutative 85.0%

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

       12. distribute-rgt-neg-in 85.0%

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

       13. distribute-rgt-in 85.0%

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

       14. fma-define 85.0%

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

       15. distribute-lft-neg-in 85.0%

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

       16. sub-neg 85.0%

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

       17. *-commutative 85.0%

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

   14. Simplified 85.0%

       \[\leadsto \color{blue}{\ell \cdot \left(1 - w \cdot \left(1 + \log \ell\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell + \left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{elif}\;\ell \leq 10^{+39}:\\ \;\;\;\;\frac{\log \ell \cdot \left(w \cdot \left(-\ell\right)\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 - \log \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 - w\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (- 1.0 w)))

C

double code(double w, double l) {
	return l * (1.0 - w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 - w)
end function

Java

public static double code(double w, double l) {
	return l * (1.0 - w);
}

Python

def code(w, l):
	return l * (1.0 - w)

Julia

function code(w, l)
	return Float64(l * Float64(1.0 - w))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 - w);
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in w around 0 59.0%

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

   1. add-log-exp 32.1%

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

   2. *-commutative 32.1%

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

   3. exp-to-pow 32.1%

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

7. Applied egg-rr 32.1%

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

8. Taylor expanded in l around inf 68.4%

   \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 68.4%

      \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

   2. unsub-neg 68.4%

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

10. Simplified 68.4%

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

11. Final simplification 68.4%

    \[\leadsto \ell \cdot \left(1 - w\right) \]
12. Add Preprocessing

Alternative 7: 57.3% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in w around 0 59.8%

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

6. Final simplification 59.8%

   \[\leadsto \ell \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))