exp-w (used to crash)

Percentage Accurate: 99.6% → 99.6%

Time: 11.3s

Alternatives: 13

Speedup: 1.0×

• 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: 99.6% 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: 99.6% 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]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.6% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

Alternative 3: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 20000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 20000.0))) (exp (- w)) (* l (exp w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 20000.0)) {
		tmp = exp(-w);
	} else {
		tmp = 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 <= (-0.7d0)) .or. (.not. (w <= 20000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l * exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 20000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l * Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 20000.0):
		tmp = math.exp(-w)
	else:
		tmp = l * math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 20000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l * exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 20000.0)))
		tmp = exp(-w);
	else
		tmp = l * exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 20000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l * N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 2e4 < 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. 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. Step-by-step derivation

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 63.4%

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

      3. sqr-neg 63.4%

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

      4. sqrt-unprod 37.5%

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

      5. add-sqr-sqrt 63.4%

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

      6. add-sqr-sqrt 63.4%

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

      7. sqrt-unprod 63.4%

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

      8. add-sqr-sqrt 37.5%

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

      9. sqrt-unprod 37.5%

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

      10. sqr-neg 37.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 99.2%

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

      17. metadata-eval 99.2%

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

      18. metadata-eval 99.2%

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

      19. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]
   7. Step-by-step derivation

      1. add-exp-log 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. log-div 99.2%

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

      4. add-log-exp 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   10. Step-by-step derivation

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -0.69999999999999996 < w < 2e4

   1. Initial program 98.8%

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

      1. exp-neg 98.8%

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

      2. remove-double-neg 98.8%

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

      3. associate-*l/ 98.8%

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

      4. *-lft-identity 98.8%

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

      5. remove-double-neg 98.8%

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

   3. Simplified 98.8%

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

      1. add-sqr-sqrt 46.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 51.0%

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

      5. add-sqr-sqrt 95.5%

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

      6. add-sqr-sqrt 95.5%

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

      7. sqrt-unprod 95.5%

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

      8. add-sqr-sqrt 51.0%

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

      9. sqrt-unprod 95.5%

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

      10. sqr-neg 95.5%

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

      11. sqrt-unprod 44.6%

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

      12. add-sqr-sqrt 95.6%

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

      13. pow1 95.6%

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

      14. exp-neg 95.6%

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

      15. inv-pow 95.6%

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

      16. pow-prod-up 95.6%

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

      17. metadata-eval 95.6%

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

      18. metadata-eval 95.6%

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

      19. metadata-eval 95.6%

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

   6. Applied egg-rr 95.6%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]
   7. Step-by-step derivation

      1. associate-/l* 95.6%

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

      2. *-commutative 95.6%

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

      3. rec-exp 95.6%

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

   8. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{e^{-w} \cdot \ell} \]
   9. Step-by-step derivation

      1. pow1 95.6%

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

      2. *-commutative 95.6%

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

      3. add-sqr-sqrt 51.0%

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

      4. sqrt-unprod 96.3%

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

      5. sqr-neg 96.3%

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

      6. sqrt-unprod 45.3%

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

      7. add-sqr-sqrt 96.3%

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

   10. Applied egg-rr 96.3%

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

       1. unpow1 96.3%

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

   12. Simplified 96.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 20000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.72 \lor \neg \left(w \leq 19000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \left(w \cdot \ell\right) \cdot \left(w \cdot -0.5 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.72) (not (<= w 19000.0)))
   (exp (- w))
   (- l (* (* w l) (+ (* w -0.5) -1.0)))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.72) || !(w <= 19000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l - ((w * l) * ((w * -0.5) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.72d0)) .or. (.not. (w <= 19000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l - ((w * l) * ((w * (-0.5d0)) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.72) || !(w <= 19000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l - ((w * l) * ((w * -0.5) + -1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.72) or not (w <= 19000.0):
		tmp = math.exp(-w)
	else:
		tmp = l - ((w * l) * ((w * -0.5) + -1.0))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.72) || !(w <= 19000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l - Float64(Float64(w * l) * Float64(Float64(w * -0.5) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.72) || ~((w <= 19000.0)))
		tmp = exp(-w);
	else
		tmp = l - ((w * l) * ((w * -0.5) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.72], N[Not[LessEqual[w, 19000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l - N[(N[(w * l), $MachinePrecision] * N[(N[(w * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.71999999999999997 or 19000 < 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. 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. Step-by-step derivation

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 63.4%

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

      3. sqr-neg 63.4%

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

      4. sqrt-unprod 37.5%

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

      5. add-sqr-sqrt 63.4%

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

      6. add-sqr-sqrt 63.4%

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

      7. sqrt-unprod 63.4%

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

      8. add-sqr-sqrt 37.5%

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

      9. sqrt-unprod 37.5%

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

      10. sqr-neg 37.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 99.2%

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

      17. metadata-eval 99.2%

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

      18. metadata-eval 99.2%

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

      19. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]
   7. Step-by-step derivation

      1. add-exp-log 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. log-div 99.2%

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

      4. add-log-exp 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   10. Step-by-step derivation

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -0.71999999999999997 < w < 19000

   1. Initial program 98.8%

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

      1. exp-neg 98.8%

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

      2. remove-double-neg 98.8%

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

      3. associate-*l/ 98.8%

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

      4. *-lft-identity 98.8%

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

      5. remove-double-neg 98.8%

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

   3. Simplified 98.8%

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

      1. add-sqr-sqrt 46.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 51.0%

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

      5. add-sqr-sqrt 95.5%

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

      6. add-sqr-sqrt 95.5%

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

      7. sqrt-unprod 95.5%

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

      8. add-sqr-sqrt 51.0%

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

      9. sqrt-unprod 95.5%

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

      10. sqr-neg 95.5%

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

      11. sqrt-unprod 44.6%

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

      12. add-sqr-sqrt 95.6%

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

      13. pow1 95.6%

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

      14. exp-neg 95.6%

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

      15. inv-pow 95.6%

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

      16. pow-prod-up 95.6%

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

      17. metadata-eval 95.6%

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

      18. metadata-eval 95.6%

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

      19. metadata-eval 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in w around 0 95.6%

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 95.6%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-1 \cdot w\right) \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)} - \ell\right) \]

      2. neg-mul-1 95.6%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-w\right)} \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) - \ell\right) \]

      3. distribute-rgt-out 95.6%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \color{blue}{\left(\ell \cdot \left(-1 + 0.5\right)\right)} - \ell\right) \]

      4. metadata-eval 95.6%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot \color{blue}{-0.5}\right) - \ell\right) \]

   9. Simplified 95.6%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 44.6%

          \[\leadsto \ell + \color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)} \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right) \]

       2. sqrt-unprod 95.6%

          \[\leadsto \ell + \color{blue}{\sqrt{w \cdot w}} \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right) \]

       3. sqr-neg 95.6%

          \[\leadsto \ell + \sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}} \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right) \]

       4. sqrt-unprod 51.0%

          \[\leadsto \ell + \color{blue}{\left(\sqrt{-w} \cdot \sqrt{-w}\right)} \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right) \]

       5. add-sqr-sqrt 95.6%

          \[\leadsto \ell + \color{blue}{\left(-w\right)} \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right) \]

       6. cancel-sign-sub-inv 95.6%

          \[\leadsto \color{blue}{\ell - w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right)} \]

       7. *-commutative 95.6%

          \[\leadsto \ell - w \cdot \left(\color{blue}{\left(\ell \cdot -0.5\right) \cdot \left(-w\right)} - \ell\right) \]

       8. associate-*l* 95.6%

          \[\leadsto \ell - w \cdot \left(\color{blue}{\ell \cdot \left(-0.5 \cdot \left(-w\right)\right)} - \ell\right) \]

       9. add-sqr-sqrt 51.0%

          \[\leadsto \ell - w \cdot \left(\ell \cdot \left(-0.5 \cdot \color{blue}{\left(\sqrt{-w} \cdot \sqrt{-w}\right)}\right) - \ell\right) \]

       10. sqrt-unprod 95.6%

           \[\leadsto \ell - w \cdot \left(\ell \cdot \left(-0.5 \cdot \color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}\right) - \ell\right) \]

       11. sqr-neg 95.6%

           \[\leadsto \ell - w \cdot \left(\ell \cdot \left(-0.5 \cdot \sqrt{\color{blue}{w \cdot w}}\right) - \ell\right) \]

       12. sqrt-unprod 44.6%

           \[\leadsto \ell - w \cdot \left(\ell \cdot \left(-0.5 \cdot \color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)}\right) - \ell\right) \]

       13. add-sqr-sqrt 95.6%

           \[\leadsto \ell - w \cdot \left(\ell \cdot \left(-0.5 \cdot \color{blue}{w}\right) - \ell\right) \]

   11. Applied egg-rr 95.6%

       \[\leadsto \color{blue}{\ell - w \cdot \left(\ell \cdot \left(-0.5 \cdot w\right) - \ell\right)} \]
   12. Step-by-step derivation

       1. sub-neg 95.6%

          \[\leadsto \ell - w \cdot \color{blue}{\left(\ell \cdot \left(-0.5 \cdot w\right) + \left(-\ell\right)\right)} \]

       2. distribute-lft-out 95.6%

          \[\leadsto \ell - \color{blue}{\left(w \cdot \left(\ell \cdot \left(-0.5 \cdot w\right)\right) + w \cdot \left(-\ell\right)\right)} \]

       3. associate-*r* 95.6%

          \[\leadsto \ell - \left(\color{blue}{\left(w \cdot \ell\right) \cdot \left(-0.5 \cdot w\right)} + w \cdot \left(-\ell\right)\right) \]

       4. distribute-rgt-neg-out 95.6%

          \[\leadsto \ell - \left(\left(w \cdot \ell\right) \cdot \left(-0.5 \cdot w\right) + \color{blue}{\left(-w \cdot \ell\right)}\right) \]

       5. neg-mul-1 95.6%

          \[\leadsto \ell - \left(\left(w \cdot \ell\right) \cdot \left(-0.5 \cdot w\right) + \color{blue}{-1 \cdot \left(w \cdot \ell\right)}\right) \]

       6. *-commutative 95.6%

          \[\leadsto \ell - \left(\left(w \cdot \ell\right) \cdot \left(-0.5 \cdot w\right) + \color{blue}{\left(w \cdot \ell\right) \cdot -1}\right) \]

       7. distribute-lft-out 95.6%

          \[\leadsto \ell - \color{blue}{\left(w \cdot \ell\right) \cdot \left(-0.5 \cdot w + -1\right)} \]

       8. *-commutative 95.6%

          \[\leadsto \ell - \color{blue}{\left(\ell \cdot w\right)} \cdot \left(-0.5 \cdot w + -1\right) \]

       9. *-commutative 95.6%

          \[\leadsto \ell - \left(\ell \cdot w\right) \cdot \left(\color{blue}{w \cdot -0.5} + -1\right) \]

   13. Simplified 95.6%

       \[\leadsto \color{blue}{\ell - \left(\ell \cdot w\right) \cdot \left(w \cdot -0.5 + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.72 \lor \neg \left(w \leq 19000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \left(w \cdot \ell\right) \cdot \left(w \cdot -0.5 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{e^{w}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l / math.exp(w)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 37.7%

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

   2. sqrt-unprod 82.8%

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

   3. sqr-neg 82.8%

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

   4. sqrt-unprod 45.1%

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

   5. add-sqr-sqrt 81.5%

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

   6. add-sqr-sqrt 81.5%

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

   7. sqrt-unprod 81.5%

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

   8. add-sqr-sqrt 45.1%

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

   9. sqrt-unprod 70.2%

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

   10. sqr-neg 70.2%

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

   11. sqrt-unprod 25.1%

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

   12. add-sqr-sqrt 53.8%

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

   13. pow1 53.8%

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

   14. exp-neg 53.8%

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

   15. inv-pow 53.8%

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

   16. pow-prod-up 97.2%

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

   17. metadata-eval 97.2%

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

   18. metadata-eval 97.2%

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

   19. metadata-eval 97.2%

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

6. Applied egg-rr 97.2%

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

7. Taylor expanded in l around 0 97.2%

   \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]
8. Add Preprocessing

Alternative 6: 84.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.55e-30)
   (+ l (* l (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666)))))))
   (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.55e-30) {
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = w * (l / 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 <= 1.55d-30) then
        tmp = l + (l * (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0)))))))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 1.55e-30) {
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.55e-30:
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.55e-30)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.55e-30)
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.55e-30], N[(l + N[(l * N[(w * N[(-1.0 + N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.54999999999999995e-30

   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. Step-by-step derivation

      1. add-sqr-sqrt 27.9%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. sqrt-unprod 52.7%

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

      5. add-sqr-sqrt 80.6%

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

      6. add-sqr-sqrt 80.6%

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

      7. sqrt-unprod 80.6%

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

      8. add-sqr-sqrt 52.7%

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

      9. sqrt-unprod 80.6%

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

      10. sqr-neg 80.6%

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

      11. sqrt-unprod 27.9%

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

      12. add-sqr-sqrt 61.4%

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

      13. pow1 61.4%

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

      14. exp-neg 61.4%

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

      15. inv-pow 61.4%

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

      16. pow-prod-up 98.9%

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

      17. metadata-eval 98.9%

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

      18. metadata-eval 98.9%

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

      19. metadata-eval 98.9%

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

   6. Applied egg-rr 98.9%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]
   7. Step-by-step derivation

      1. associate-/l* 98.9%

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

      2. *-commutative 98.9%

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

      3. rec-exp 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in w around 0 86.1%

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \ell + w \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot w\right) + 0.5 \cdot \ell\right)\right)} \]

   10. Taylor expanded in l around 0 87.8%

       \[\leadsto \ell + \color{blue}{\ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right)} \]

   if 1.54999999999999995e-30 < w

   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. remove-double-neg 96.1%

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

      3. associate-*l/ 96.1%

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

      4. *-lft-identity 96.1%

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

      5. remove-double-neg 96.1%

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

   3. Simplified 96.1%

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

      1. add-sqr-sqrt 96.1%

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

      2. sqrt-unprod 96.1%

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

      3. sqr-neg 96.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 86.9%

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

      6. add-sqr-sqrt 86.9%

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

      7. sqrt-unprod 86.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 8.7%

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

      10. sqr-neg 8.7%

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

      11. sqrt-unprod 8.7%

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

      12. add-sqr-sqrt 8.7%

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

      13. pow1 8.7%

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

      14. exp-neg 8.7%

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

      15. inv-pow 8.7%

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

      16. pow-prod-up 87.1%

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

      17. metadata-eval 87.1%

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

      18. metadata-eval 87.1%

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

      19. metadata-eval 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in w around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. unsub-neg 11.3%

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

      3. *-rgt-identity 11.3%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 11.3%

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

   9. Simplified 11.3%

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

   10. Taylor expanded in w around inf 11.3%

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

       1. neg-mul-1 11.3%

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

       2. +-commutative 11.3%

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

       3. sub-neg 11.3%

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

   12. Simplified 11.3%

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

   13. Taylor expanded in w around 0 48.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 54000:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 54000.0) (+ l (* l (* w (+ -1.0 (* w 0.5))))) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 54000.0) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = w * (l / 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 <= 54000.0d0) then
        tmp = l + (l * (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 54000.0) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 54000.0:
		tmp = l + (l * (w * (-1.0 + (w * 0.5))))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 54000.0)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 54000.0)
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 54000.0], N[(l + N[(l * N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 54000

   1. Initial program 99.3%

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

      1. exp-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. associate-*l/ 99.3%

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

      4. *-lft-identity 99.3%

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

      5. remove-double-neg 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 29.8%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. sqrt-unprod 50.8%

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

      5. add-sqr-sqrt 79.1%

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

      6. add-sqr-sqrt 79.1%

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

      7. sqrt-unprod 79.1%

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

      8. add-sqr-sqrt 50.8%

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

      9. sqrt-unprod 79.1%

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

      10. sqr-neg 79.1%

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

      11. sqrt-unprod 28.3%

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

      12. add-sqr-sqrt 60.7%

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

      13. pow1 60.7%

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

      14. exp-neg 60.7%

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

      15. inv-pow 60.7%

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

      16. pow-prod-up 96.8%

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

      17. metadata-eval 96.8%

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

      18. metadata-eval 96.8%

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

      19. metadata-eval 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in w around 0 80.7%

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 80.7%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-1 \cdot w\right) \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)} - \ell\right) \]

      2. neg-mul-1 80.7%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-w\right)} \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) - \ell\right) \]

      3. distribute-rgt-out 80.7%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \color{blue}{\left(\ell \cdot \left(-1 + 0.5\right)\right)} - \ell\right) \]

      4. metadata-eval 80.7%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot \color{blue}{-0.5}\right) - \ell\right) \]

   9. Simplified 80.7%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right)} \]

   10. Taylor expanded in l around 0 82.8%

       \[\leadsto \ell + \color{blue}{\ell \cdot \left(w \cdot \left(0.5 \cdot w - 1\right)\right)} \]

   if 54000 < 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. 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

      3. *-rgt-identity 3.5%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. +-commutative 3.5%

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

       3. sub-neg 3.5%

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

   12. Simplified 3.5%

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

   13. Taylor expanded in w around 0 50.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 54000:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 90000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 90000.0) (* l (+ 1.0 (* w (+ -1.0 (* w 0.5))))) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 90000.0) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = w * (l / 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 <= 90000.0d0) then
        tmp = l * (1.0d0 + (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 90000.0) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 90000.0:
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 90000.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 90000.0)
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 90000.0], N[(l * N[(1.0 + N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 9e4

   1. Initial program 99.3%

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

      1. exp-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. associate-*l/ 99.3%

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

      4. *-lft-identity 99.3%

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

      5. remove-double-neg 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 29.8%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. sqrt-unprod 50.8%

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

      5. add-sqr-sqrt 79.1%

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

      6. add-sqr-sqrt 79.1%

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

      7. sqrt-unprod 79.1%

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

      8. add-sqr-sqrt 50.8%

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

      9. sqrt-unprod 79.1%

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

      10. sqr-neg 79.1%

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

      11. sqrt-unprod 28.3%

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

      12. add-sqr-sqrt 60.7%

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

      13. pow1 60.7%

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

      14. exp-neg 60.7%

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

      15. inv-pow 60.7%

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

      16. pow-prod-up 96.8%

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

      17. metadata-eval 96.8%

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

      18. metadata-eval 96.8%

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

      19. metadata-eval 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in w around 0 80.7%

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 80.7%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-1 \cdot w\right) \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)} - \ell\right) \]

      2. neg-mul-1 80.7%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-w\right)} \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) - \ell\right) \]

      3. distribute-rgt-out 80.7%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \color{blue}{\left(\ell \cdot \left(-1 + 0.5\right)\right)} - \ell\right) \]

      4. metadata-eval 80.7%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot \color{blue}{-0.5}\right) - \ell\right) \]

   9. Simplified 80.7%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right)} \]

   10. Taylor expanded in l around 0 82.8%

       \[\leadsto \color{blue}{\ell \cdot \left(1 + w \cdot \left(0.5 \cdot w - 1\right)\right)} \]

   if 9e4 < 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. 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

      3. *-rgt-identity 3.5%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. +-commutative 3.5%

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

       3. sub-neg 3.5%

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

   12. Simplified 3.5%

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

   13. Taylor expanded in w around 0 50.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 90000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 240000:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 240000.0) (+ l (* w (* l (* w 0.5)))) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 240000.0) {
		tmp = l + (w * (l * (w * 0.5)));
	} else {
		tmp = w * (l / 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 <= 240000.0d0) then
        tmp = l + (w * (l * (w * 0.5d0)))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 240000.0) {
		tmp = l + (w * (l * (w * 0.5)));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 240000.0:
		tmp = l + (w * (l * (w * 0.5)))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 240000.0)
		tmp = Float64(l + Float64(w * Float64(l * Float64(w * 0.5))));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 240000.0)
		tmp = l + (w * (l * (w * 0.5)));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 240000.0], N[(l + N[(w * N[(l * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.4e5

   1. Initial program 99.3%

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

      1. exp-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. associate-*l/ 99.3%

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

      4. *-lft-identity 99.3%

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

      5. remove-double-neg 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 29.8%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. sqrt-unprod 50.8%

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

      5. add-sqr-sqrt 79.1%

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

      6. add-sqr-sqrt 79.1%

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

      7. sqrt-unprod 79.1%

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

      8. add-sqr-sqrt 50.8%

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

      9. sqrt-unprod 79.1%

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

      10. sqr-neg 79.1%

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

      11. sqrt-unprod 28.3%

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

      12. add-sqr-sqrt 60.7%

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

      13. pow1 60.7%

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

      14. exp-neg 60.7%

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

      15. inv-pow 60.7%

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

      16. pow-prod-up 96.8%

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

      17. metadata-eval 96.8%

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

      18. metadata-eval 96.8%

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

      19. metadata-eval 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in w around 0 80.7%

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 80.7%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-1 \cdot w\right) \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)} - \ell\right) \]

      2. neg-mul-1 80.7%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-w\right)} \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) - \ell\right) \]

      3. distribute-rgt-out 80.7%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \color{blue}{\left(\ell \cdot \left(-1 + 0.5\right)\right)} - \ell\right) \]

      4. metadata-eval 80.7%

         \[\leadsto \ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot \color{blue}{-0.5}\right) - \ell\right) \]

   9. Simplified 80.7%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\left(-w\right) \cdot \left(\ell \cdot -0.5\right) - \ell\right)} \]

   10. Taylor expanded in w around inf 80.7%

       \[\leadsto \ell + w \cdot \color{blue}{\left(0.5 \cdot \left(\ell \cdot w\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 80.7%

          \[\leadsto \ell + w \cdot \color{blue}{\left(\left(0.5 \cdot \ell\right) \cdot w\right)} \]

       2. *-commutative 80.7%

          \[\leadsto \ell + w \cdot \left(\color{blue}{\left(\ell \cdot 0.5\right)} \cdot w\right) \]

       3. associate-*l* 80.7%

          \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \left(0.5 \cdot w\right)\right)} \]

       4. *-commutative 80.7%

          \[\leadsto \ell + w \cdot \left(\ell \cdot \color{blue}{\left(w \cdot 0.5\right)}\right) \]

   12. Simplified 80.7%

       \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \left(w \cdot 0.5\right)\right)} \]

   if 2.4e5 < 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. 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

      3. *-rgt-identity 3.5%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. +-commutative 3.5%

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

       3. sub-neg 3.5%

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

   12. Simplified 3.5%

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

   13. Taylor expanded in w around 0 50.7%

       \[\leadsto w \cdot \color{blue}{\frac{\ell}{w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.2% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.55e-30) (* l (- 1.0 w)) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.55e-30) {
		tmp = l * (1.0 - w);
	} else {
		tmp = w * (l / 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 <= 1.55d-30) then
        tmp = l * (1.0d0 - w)
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 1.55e-30) {
		tmp = l * (1.0 - w);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.55e-30:
		tmp = l * (1.0 - w)
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.55e-30)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.55e-30)
		tmp = l * (1.0 - w);
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.55e-30], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.54999999999999995e-30

   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. Step-by-step derivation

      1. add-sqr-sqrt 27.9%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. sqrt-unprod 52.7%

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

      5. add-sqr-sqrt 80.6%

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

      6. add-sqr-sqrt 80.6%

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

      7. sqrt-unprod 80.6%

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

      8. add-sqr-sqrt 52.7%

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

      9. sqrt-unprod 80.6%

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

      10. sqr-neg 80.6%

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

      11. sqrt-unprod 27.9%

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

      12. add-sqr-sqrt 61.4%

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

      13. pow1 61.4%

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

      14. exp-neg 61.4%

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

      15. inv-pow 61.4%

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

      16. pow-prod-up 98.9%

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

      17. metadata-eval 98.9%

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

      18. metadata-eval 98.9%

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

      19. metadata-eval 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in w around 0 71.9%

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. *-rgt-identity 71.9%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 71.9%

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

   9. Simplified 71.9%

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

   if 1.54999999999999995e-30 < w

   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. remove-double-neg 96.1%

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

      3. associate-*l/ 96.1%

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

      4. *-lft-identity 96.1%

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

      5. remove-double-neg 96.1%

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

   3. Simplified 96.1%

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

      1. add-sqr-sqrt 96.1%

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

      2. sqrt-unprod 96.1%

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

      3. sqr-neg 96.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 86.9%

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

      6. add-sqr-sqrt 86.9%

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

      7. sqrt-unprod 86.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 8.7%

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

      10. sqr-neg 8.7%

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

      11. sqrt-unprod 8.7%

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

      12. add-sqr-sqrt 8.7%

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

      13. pow1 8.7%

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

      14. exp-neg 8.7%

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

      15. inv-pow 8.7%

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

      16. pow-prod-up 87.1%

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

      17. metadata-eval 87.1%

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

      18. metadata-eval 87.1%

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

      19. metadata-eval 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in w around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. unsub-neg 11.3%

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

      3. *-rgt-identity 11.3%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 11.3%

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

   9. Simplified 11.3%

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

   10. Taylor expanded in w around inf 11.3%

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

       1. neg-mul-1 11.3%

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

       2. +-commutative 11.3%

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

       3. sub-neg 11.3%

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

   12. Simplified 11.3%

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

   13. Taylor expanded in w around 0 48.6%

       \[\leadsto w \cdot \color{blue}{\frac{\ell}{w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.5% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.0125:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.0125) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.0125) {
		tmp = w * -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 <= (-0.0125d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.0125) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.0125:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.0125)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.0125)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.0125], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.012500000000000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 50.6%

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

      3. sqr-neg 50.6%

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

      4. sqrt-unprod 50.6%

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

      5. add-sqr-sqrt 50.6%

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

      6. add-sqr-sqrt 50.6%

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

      7. sqrt-unprod 50.6%

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

      8. add-sqr-sqrt 50.6%

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

      9. sqrt-unprod 50.6%

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

      10. sqr-neg 50.6%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.9%

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

      17. metadata-eval 98.9%

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

      18. metadata-eval 98.9%

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

      19. metadata-eval 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in w around 0 27.8%

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

      1. mul-1-neg 27.8%

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

      2. unsub-neg 27.8%

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

      3. *-rgt-identity 27.8%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in w around inf 27.8%

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

       1. mul-1-neg 27.8%

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

       2. distribute-lft-neg-out 27.8%

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

       3. *-commutative 27.8%

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

   12. Simplified 27.8%

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

   if -0.012500000000000001 < w

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 80.5%

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

Alternative 12: 64.2% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 37.7%

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

   2. sqrt-unprod 82.8%

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

   3. sqr-neg 82.8%

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

   4. sqrt-unprod 45.1%

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

   5. add-sqr-sqrt 81.5%

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

   6. add-sqr-sqrt 81.5%

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

   7. sqrt-unprod 81.5%

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

   8. add-sqr-sqrt 45.1%

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

   9. sqrt-unprod 70.2%

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

   10. sqr-neg 70.2%

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

   11. sqrt-unprod 25.1%

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

   12. add-sqr-sqrt 53.8%

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

   13. pow1 53.8%

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

   14. exp-neg 53.8%

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

   15. inv-pow 53.8%

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

   16. pow-prod-up 97.2%

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

   17. metadata-eval 97.2%

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

   18. metadata-eval 97.2%

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

   19. metadata-eval 97.2%

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

6. Applied egg-rr 97.2%

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

7. Taylor expanded in w around 0 63.1%

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

   1. mul-1-neg 63.1%

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

   2. unsub-neg 63.1%

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

   3. *-rgt-identity 63.1%

      \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

   4. distribute-lft-out-- 63.1%

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

9. Simplified 63.1%

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

Alternative 13: 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 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Taylor expanded in w around 0 55.6%

   \[\leadsto \color{blue}{\ell} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))