exp-w (used to crash)

Percentage Accurate: 99.4% → 99.1%

Time: 19.5s

Alternatives: 20

Speedup: 1.0×

• could not determine a ground truth

  1. w = 7.9989231020831345e-289
  2. l = -1.3113663863915876e-13

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -290:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -290.0)
   (/ l (exp w))
   (/ (pow l (exp w)) (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -290.0) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-290.0d0)) then
        tmp = l / exp(w)
    else
        tmp = (l ** exp(w)) / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -290.0) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -290.0:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -290.0)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ exp(w)) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -290.0)
		tmp = l / exp(w);
	else
		tmp = (l ^ exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -290.0], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -290

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 46.9%

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

      5. add-sqr-sqrt 46.9%

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

      6. add-sqr-sqrt 46.9%

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

      7. sqrt-unprod 46.9%

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

      8. add-sqr-sqrt 46.9%

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

      9. sqrt-unprod 46.9%

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

      10. sqr-neg 46.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   if -290 < w

   1. Initial program 98.5%

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

      1. exp-neg 98.5%

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

      2. remove-double-neg 98.5%

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

      3. associate-*l/ 98.5%

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

      4. *-lft-identity 98.5%

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

      5. remove-double-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in w around 0 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + w \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

Alternative 4: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) * (1.0 / (w + 1.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.2%

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

      3. sqr-neg 46.2%

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

      4. sqrt-unprod 46.2%

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

      5. add-sqr-sqrt 46.2%

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

      6. add-sqr-sqrt 46.2%

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

      7. sqrt-unprod 46.2%

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

      8. add-sqr-sqrt 46.2%

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

      9. sqrt-unprod 46.2%

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

      10. sqr-neg 46.2%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.5%

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

      17. metadata-eval 98.5%

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

      18. metadata-eval 98.5%

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

      19. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in w around inf 98.5%

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

      1. exp-neg 98.5%

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

      2. associate-*r/ 98.5%

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

      3. *-rgt-identity 98.5%

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

   7. Simplified 98.5%

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

   if -1 < w

   1. Initial program 98.5%

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

      1. exp-neg 98.5%

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

      2. remove-double-neg 98.5%

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

      3. associate-*l/ 98.5%

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

      4. *-lft-identity 98.5%

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

      5. remove-double-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. div-inv 98.8%

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

   9. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\ell}^{\left(e^{w}\right)} \cdot \frac{1}{w + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0) (/ l (exp w)) (/ (pow l (exp w)) (+ w 1.0))))

C

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

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) / (w + 1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.2%

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

      3. sqr-neg 46.2%

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

      4. sqrt-unprod 46.2%

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

      5. add-sqr-sqrt 46.2%

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

      6. add-sqr-sqrt 46.2%

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

      7. sqrt-unprod 46.2%

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

      8. add-sqr-sqrt 46.2%

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

      9. sqrt-unprod 46.2%

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

      10. sqr-neg 46.2%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.5%

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

      17. metadata-eval 98.5%

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

      18. metadata-eval 98.5%

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

      19. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in w around inf 98.5%

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

      1. exp-neg 98.5%

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

      2. associate-*r/ 98.5%

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

      3. *-rgt-identity 98.5%

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

   7. Simplified 98.5%

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

   if -1 < w

   1. Initial program 98.5%

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

      1. exp-neg 98.5%

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

      2. remove-double-neg 98.5%

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

      3. associate-*l/ 98.5%

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

      4. *-lft-identity 98.5%

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

      5. remove-double-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. add-sqr-sqrt 42.0%

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

   2. sqrt-unprod 84.2%

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

   3. sqr-neg 84.2%

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

   4. sqrt-unprod 42.1%

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

   5. add-sqr-sqrt 83.5%

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

   6. add-sqr-sqrt 83.5%

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

   7. sqrt-unprod 83.5%

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

   8. add-sqr-sqrt 42.1%

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

   9. sqrt-unprod 60.9%

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

   10. sqr-neg 60.9%

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

   11. sqrt-unprod 18.8%

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

   12. add-sqr-sqrt 49.2%

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

   13. pow1 49.2%

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

   14. exp-neg 49.2%

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

   15. inv-pow 49.2%

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

   16. pow-prod-up 96.9%

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

   17. metadata-eval 96.9%

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

   18. metadata-eval 96.9%

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

   19. metadata-eval 96.9%

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

4. Applied egg-rr 96.9%

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

5. Final simplification 96.9%

   \[\leadsto e^{-w} \cdot \ell \]
6. Add Preprocessing

Alternative 7: 97.7% 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 98.9%

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

   1. add-sqr-sqrt 42.0%

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

   2. sqrt-unprod 84.2%

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

   3. sqr-neg 84.2%

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

   4. sqrt-unprod 42.1%

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

   5. add-sqr-sqrt 83.5%

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

   6. add-sqr-sqrt 83.5%

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

   7. sqrt-unprod 83.5%

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

   8. add-sqr-sqrt 42.1%

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

   9. sqrt-unprod 60.9%

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

   10. sqr-neg 60.9%

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

   11. sqrt-unprod 18.8%

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

   12. add-sqr-sqrt 49.2%

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

   13. pow1 49.2%

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

   14. exp-neg 49.2%

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

   15. inv-pow 49.2%

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

   16. pow-prod-up 96.9%

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

   17. metadata-eval 96.9%

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

   18. metadata-eval 96.9%

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

   19. metadata-eval 96.9%

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

4. Applied egg-rr 96.9%

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

5. Taylor expanded in w around inf 96.9%

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

   1. exp-neg 96.9%

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

   2. associate-*r/ 96.9%

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

   3. *-rgt-identity 96.9%

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

7. Simplified 96.9%

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

Alternative 8: 91.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;w \leq 0.078:\\ \;\;\;\;\ell \cdot t\_0\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{+97}:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0)))))
   (if (<= w 0.078)
     (* l t_0)
     (if (<= w 2.5e+97)
       (* t_0 0.0)
       (/ l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))))

C

double code(double w, double l) {
	double t_0 = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0));
	double tmp;
	if (w <= 0.078) {
		tmp = l * t_0;
	} else if (w <= 2.5e+97) {
		tmp = t_0 * 0.0;
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0)))
    if (w <= 0.078d0) then
        tmp = l * t_0
    else if (w <= 2.5d+97) then
        tmp = t_0 * 0.0d0
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0));
	double tmp;
	if (w <= 0.078) {
		tmp = l * t_0;
	} else if (w <= 2.5e+97) {
		tmp = t_0 * 0.0;
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0))
	tmp = 0
	if w <= 0.078:
		tmp = l * t_0
	elif w <= 2.5e+97:
		tmp = t_0 * 0.0
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	t_0 = Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0)))
	tmp = 0.0
	if (w <= 0.078)
		tmp = Float64(l * t_0);
	elseif (w <= 2.5e+97)
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0));
	tmp = 0.0;
	if (w <= 0.078)
		tmp = l * t_0;
	elseif (w <= 2.5e+97)
		tmp = t_0 * 0.0;
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 0.078], N[(l * t$95$0), $MachinePrecision], If[LessEqual[w, 2.5e+97], N[(t$95$0 * 0.0), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if w < 0.0779999999999999999

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 24.9%

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

      2. sqrt-unprod 80.2%

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

      3. sqr-neg 80.2%

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

      4. sqrt-unprod 55.3%

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

      5. add-sqr-sqrt 79.9%

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

      6. add-sqr-sqrt 79.9%

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

      7. sqrt-unprod 79.9%

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

      8. add-sqr-sqrt 55.3%

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

      9. sqrt-unprod 79.9%

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

      10. sqr-neg 79.9%

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

      11. sqrt-unprod 24.6%

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

      12. add-sqr-sqrt 64.6%

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

      13. pow1 64.6%

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

      14. exp-neg 64.6%

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

      15. inv-pow 64.6%

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

      16. pow-prod-up 97.4%

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

      17. metadata-eval 97.4%

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

      18. metadata-eval 97.4%

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

      19. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in w around 0 87.1%

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

   6. Taylor expanded in l around 0 87.1%

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

   if 0.0779999999999999999 < w < 2.49999999999999999e97

   1. Initial program 91.7%

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

      1. add-sqr-sqrt 91.7%

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

      2. sqrt-unprod 91.7%

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

      3. sqr-neg 91.7%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 87.6%

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

      6. add-sqr-sqrt 87.6%

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

      7. sqrt-unprod 87.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.3%

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

      10. sqr-neg 0.3%

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

      11. sqrt-unprod 0.3%

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

      12. add-sqr-sqrt 0.3%

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

      13. pow1 0.3%

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

      14. exp-neg 0.3%

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

      15. inv-pow 0.3%

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

      16. pow-prod-up 87.9%

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

      17. metadata-eval 87.9%

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

      18. metadata-eval 87.9%

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

      19. metadata-eval 87.9%

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in w around 0 3.2%

      \[\leadsto \color{blue}{\left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right)} \cdot \left(\ell \cdot 1\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 3.2%

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

      2. expm1-log1p-u 3.2%

         \[\leadsto \left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell\right)\right)} \]

      3. expm1-undefine 79.6%

         \[\leadsto \left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\ell\right)} - 1\right)} \]

      4. log1p-undefine 79.6%

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

      5. rem-exp-log 79.6%

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

      6. +-commutative 79.6%

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

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in l around 0 95.9%

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

   if 2.49999999999999999e97 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 100.0%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + 0.16666666666666666 \cdot w\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \frac{\ell}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.078:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{+97}:\\ \;\;\;\;\left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;w \leq 2.8 \cdot 10^{+84}:\\ \;\;\;\;\left(\left(\ell + 1\right) + -1\right) \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2e-13)
   (* l (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (if (<= w 2.8e+84)
     (* (+ (+ l 1.0) -1.0) (+ 1.0 (* w (+ (* w 0.5) -1.0))))
     (/ l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666))))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2e-13) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else if (w <= 2.8e+84) {
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 2d-13) then
        tmp = l * (1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (w <= 2.8d+84) then
        tmp = ((l + 1.0d0) + (-1.0d0)) * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2e-13) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else if (w <= 2.8e+84) {
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2e-13:
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	elif w <= 2.8e+84:
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2e-13)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	elseif (w <= 2.8e+84)
		tmp = Float64(Float64(Float64(l + 1.0) + -1.0) * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2e-13)
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	elseif (w <= 2.8e+84)
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2e-13], N[(l * N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.8e+84], N[(N[(N[(l + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < 2.0000000000000001e-13

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 24.3%

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

      2. sqrt-unprod 80.2%

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

      3. sqr-neg 80.2%

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

      4. sqrt-unprod 55.9%

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

      5. add-sqr-sqrt 80.1%

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

      6. add-sqr-sqrt 80.1%

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

      7. sqrt-unprod 80.1%

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

      8. add-sqr-sqrt 55.9%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 24.2%

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

      12. add-sqr-sqrt 64.6%

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

      13. pow1 64.6%

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

      14. exp-neg 64.6%

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

      15. inv-pow 64.6%

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

      16. pow-prod-up 97.8%

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

      17. metadata-eval 97.8%

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

      18. metadata-eval 97.8%

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

      19. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in w around 0 87.4%

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

   6. Taylor expanded in l around 0 87.4%

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

   if 2.0000000000000001e-13 < w < 2.79999999999999982e84

   1. Initial program 90.1%

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

      1. add-sqr-sqrt 90.1%

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

      2. sqrt-unprod 90.1%

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

      3. sqr-neg 90.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 83.8%

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

      6. add-sqr-sqrt 83.8%

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

      7. sqrt-unprod 83.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 5.9%

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

      10. sqr-neg 5.9%

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

      11. sqrt-unprod 5.9%

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

      12. add-sqr-sqrt 5.9%

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

      13. pow1 5.9%

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

      14. exp-neg 5.9%

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

      15. inv-pow 5.9%

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

      16. pow-prod-up 84.2%

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

      17. metadata-eval 84.2%

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

      18. metadata-eval 84.2%

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

      19. metadata-eval 84.2%

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

   4. Applied egg-rr 84.2%

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

   5. Taylor expanded in w around 0 8.6%

      \[\leadsto \color{blue}{\left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right)} \cdot \left(\ell \cdot 1\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 8.6%

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

      2. expm1-log1p-u 8.6%

         \[\leadsto \left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell\right)\right)} \]

      3. expm1-undefine 79.8%

         \[\leadsto \left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\ell\right)} - 1\right)} \]

      4. log1p-undefine 79.8%

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

      5. rem-exp-log 79.8%

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

      6. +-commutative 79.8%

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

   7. Applied egg-rr 79.8%

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

   8. Taylor expanded in w around 0 80.1%

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

      1. *-commutative 80.1%

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

   10. Simplified 80.1%

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

   if 2.79999999999999982e84 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 97.7%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + 0.16666666666666666 \cdot w\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 97.7%

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

   10. Simplified 97.7%

       \[\leadsto \frac{\ell}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;w \leq 2.8 \cdot 10^{+84}:\\ \;\;\;\;\left(\left(\ell + 1\right) + -1\right) \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;\left(\left(\ell + 1\right) + -1\right) \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2e-13)
   (* l (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (if (<= w 2.5e+125)
     (* (+ (+ l 1.0) -1.0) (+ 1.0 (* w (+ (* w 0.5) -1.0))))
     (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2e-13) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else if (w <= 2.5e+125) {
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 2d-13) then
        tmp = l * (1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (w <= 2.5d+125) then
        tmp = ((l + 1.0d0) + (-1.0d0)) * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2e-13) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else if (w <= 2.5e+125) {
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2e-13:
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	elif w <= 2.5e+125:
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2e-13)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	elseif (w <= 2.5e+125)
		tmp = Float64(Float64(Float64(l + 1.0) + -1.0) * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2e-13)
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	elseif (w <= 2.5e+125)
		tmp = ((l + 1.0) + -1.0) * (1.0 + (w * ((w * 0.5) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2e-13], N[(l * N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.5e+125], N[(N[(N[(l + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < 2.0000000000000001e-13

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 24.3%

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

      2. sqrt-unprod 80.2%

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

      3. sqr-neg 80.2%

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

      4. sqrt-unprod 55.9%

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

      5. add-sqr-sqrt 80.1%

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

      6. add-sqr-sqrt 80.1%

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

      7. sqrt-unprod 80.1%

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

      8. add-sqr-sqrt 55.9%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 24.2%

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

      12. add-sqr-sqrt 64.6%

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

      13. pow1 64.6%

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

      14. exp-neg 64.6%

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

      15. inv-pow 64.6%

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

      16. pow-prod-up 97.8%

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

      17. metadata-eval 97.8%

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

      18. metadata-eval 97.8%

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

      19. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in w around 0 87.4%

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

   6. Taylor expanded in l around 0 87.4%

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

   if 2.0000000000000001e-13 < w < 2.49999999999999981e125

   1. Initial program 92.9%

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

      1. add-sqr-sqrt 92.9%

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

      2. sqrt-unprod 92.9%

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

      3. sqr-neg 92.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 88.3%

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

      6. add-sqr-sqrt 88.3%

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

      7. sqrt-unprod 88.3%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 4.2%

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

      10. sqr-neg 4.2%

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

      11. sqrt-unprod 4.2%

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

      12. add-sqr-sqrt 4.2%

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

      13. pow1 4.2%

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

      14. exp-neg 4.2%

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

      15. inv-pow 4.2%

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

      16. pow-prod-up 88.7%

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

      17. metadata-eval 88.7%

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

      18. metadata-eval 88.7%

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

      19. metadata-eval 88.7%

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

   4. Applied egg-rr 88.7%

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

   5. Taylor expanded in w around 0 6.7%

      \[\leadsto \color{blue}{\left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right)} \cdot \left(\ell \cdot 1\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 6.7%

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

      2. expm1-log1p-u 6.7%

         \[\leadsto \left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell\right)\right)} \]

      3. expm1-undefine 69.9%

         \[\leadsto \left(1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\ell\right)} - 1\right)} \]

      4. log1p-undefine 69.9%

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

      5. rem-exp-log 69.9%

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

      6. +-commutative 69.9%

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

   7. Applied egg-rr 69.9%

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

   8. Taylor expanded in w around 0 82.7%

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

      1. *-commutative 82.7%

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

   10. Simplified 82.7%

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

   if 2.49999999999999981e125 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 97.1%

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

      1. *-commutative 100.0%

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

   10. Simplified 97.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;\left(\left(\ell + 1\right) + -1\right) \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -38:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -38.0)
   (* l (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -38.0) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-38.0d0)) then
        tmp = l * (1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -38.0) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -38.0:
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -38.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -38.0)
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -38.0], N[(l * N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -38

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 46.9%

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

      5. add-sqr-sqrt 46.9%

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

      6. add-sqr-sqrt 46.9%

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

      7. sqrt-unprod 46.9%

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

      8. add-sqr-sqrt 46.9%

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

      9. sqrt-unprod 46.9%

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

      10. sqr-neg 46.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0 68.7%

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

   6. Taylor expanded in l around 0 68.7%

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

   if -38 < w

   1. Initial program 98.5%

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

      1. add-sqr-sqrt 56.0%

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

      2. sqrt-unprod 96.6%

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

      3. sqr-neg 96.6%

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

      4. sqrt-unprod 40.5%

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

      5. add-sqr-sqrt 95.7%

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

      6. add-sqr-sqrt 95.7%

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

      7. sqrt-unprod 95.7%

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

      8. add-sqr-sqrt 40.5%

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

      9. sqrt-unprod 65.6%

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

      10. sqr-neg 65.6%

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

      11. sqrt-unprod 25.1%

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

      12. add-sqr-sqrt 65.6%

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

      13. pow1 65.6%

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

      14. exp-neg 65.6%

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

      15. inv-pow 65.6%

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

      16. pow-prod-up 95.9%

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

      17. metadata-eval 95.9%

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

      18. metadata-eval 95.9%

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

      19. metadata-eval 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in w around inf 95.9%

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

      1. exp-neg 95.9%

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

      2. associate-*r/ 95.9%

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

      3. *-rgt-identity 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in w around 0 87.3%

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

      1. *-commutative 98.6%

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

   10. Simplified 87.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -38:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.2% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -130:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell - w \cdot \ell\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -130.0)
   (+ l (* w (- (* w (- l (* w l))) l)))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -130.0) {
		tmp = l + (w * ((w * (l - (w * l))) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-130.0d0)) then
        tmp = l + (w * ((w * (l - (w * l))) - l))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -130.0) {
		tmp = l + (w * ((w * (l - (w * l))) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -130.0:
		tmp = l + (w * ((w * (l - (w * l))) - l))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -130.0)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(l - Float64(w * l))) - l)));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -130.0)
		tmp = l + (w * ((w * (l - (w * l))) - l));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -130.0], N[(l + N[(w * N[(N[(w * N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -130

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 46.9%

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

      5. add-sqr-sqrt 46.9%

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

      6. add-sqr-sqrt 46.9%

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

      7. sqrt-unprod 46.9%

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

      8. add-sqr-sqrt 46.9%

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

      9. sqrt-unprod 46.9%

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

      10. sqr-neg 46.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 1.0%

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

      1. +-commutative 1.0%

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

   10. Simplified 1.0%

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

   11. Taylor expanded in w around 0 59.9%

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

       1. cancel-sign-sub-inv 59.9%

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

       2. metadata-eval 59.9%

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

       3. *-lft-identity 59.9%

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

       4. +-commutative 59.9%

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

       5. associate-*r* 59.9%

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

       6. mul-1-neg 59.9%

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

   13. Simplified 59.9%

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

   if -130 < w

   1. Initial program 98.5%

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

      1. add-sqr-sqrt 56.0%

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

      2. sqrt-unprod 96.6%

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

      3. sqr-neg 96.6%

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

      4. sqrt-unprod 40.5%

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

      5. add-sqr-sqrt 95.7%

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

      6. add-sqr-sqrt 95.7%

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

      7. sqrt-unprod 95.7%

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

      8. add-sqr-sqrt 40.5%

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

      9. sqrt-unprod 65.6%

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

      10. sqr-neg 65.6%

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

      11. sqrt-unprod 25.1%

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

      12. add-sqr-sqrt 65.6%

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

      13. pow1 65.6%

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

      14. exp-neg 65.6%

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

      15. inv-pow 65.6%

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

      16. pow-prod-up 95.9%

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

      17. metadata-eval 95.9%

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

      18. metadata-eval 95.9%

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

      19. metadata-eval 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in w around inf 95.9%

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

      1. exp-neg 95.9%

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

      2. associate-*r/ 95.9%

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

      3. *-rgt-identity 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in w around 0 87.3%

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

      1. *-commutative 98.6%

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

   10. Simplified 87.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -130:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell - w \cdot \ell\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 3000000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 3000000.0)
   (* l (+ 1.0 (* w (+ (* w 0.5) -1.0))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 3000000.0) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 3000000.0d0) then
        tmp = l * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 3000000.0) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 3000000.0:
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 3000000.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 3000000.0)
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 3000000.0], N[(l * N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3e6

   1. Initial program 98.6%

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

      1. add-sqr-sqrt 25.4%

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

      2. sqrt-unprod 79.6%

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

      3. sqr-neg 79.6%

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

      4. sqrt-unprod 54.2%

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

      5. add-sqr-sqrt 78.8%

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

      6. add-sqr-sqrt 78.8%

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

      7. sqrt-unprod 78.8%

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

      8. add-sqr-sqrt 54.2%

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

      9. sqrt-unprod 78.4%

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

      10. sqr-neg 78.4%

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

      11. sqrt-unprod 24.2%

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

      12. add-sqr-sqrt 63.3%

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

      13. pow1 63.3%

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

      14. exp-neg 63.3%

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

      15. inv-pow 63.3%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in w around 0 81.1%

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

   if 3e6 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 72.9%

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

      1. *-commutative 100.0%

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

   10. Simplified 72.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3000000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2200000:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2200000.0)
   (+ l (* w (- (* w l) l)))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2200000.0) {
		tmp = l + (w * ((w * l) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 2200000.0d0) then
        tmp = l + (w * ((w * l) - l))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2200000.0) {
		tmp = l + (w * ((w * l) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2200000.0:
		tmp = l + (w * ((w * l) - l))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2200000.0)
		tmp = Float64(l + Float64(w * Float64(Float64(w * l) - l)));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2200000.0)
		tmp = l + (w * ((w * l) - l));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2200000.0], N[(l + N[(w * N[(N[(w * l), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.2e6

   1. Initial program 98.6%

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

      1. add-sqr-sqrt 25.4%

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

      2. sqrt-unprod 79.6%

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

      3. sqr-neg 79.6%

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

      4. sqrt-unprod 54.2%

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

      5. add-sqr-sqrt 78.8%

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

      6. add-sqr-sqrt 78.8%

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

      7. sqrt-unprod 78.8%

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

      8. add-sqr-sqrt 54.2%

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

      9. sqrt-unprod 78.4%

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

      10. sqr-neg 78.4%

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

      11. sqrt-unprod 24.2%

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

      12. add-sqr-sqrt 63.3%

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

      13. pow1 63.3%

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

      14. exp-neg 63.3%

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

      15. inv-pow 63.3%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in w around inf 96.0%

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

      1. exp-neg 96.0%

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

      2. associate-*r/ 96.0%

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

      3. *-rgt-identity 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in w around 0 63.7%

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

      1. +-commutative 66.6%

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

   10. Simplified 63.7%

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

   11. Taylor expanded in w around 0 77.3%

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

   if 2.2e6 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 72.9%

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

      1. *-commutative 100.0%

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

   10. Simplified 72.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2200000:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.2% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 70000:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 70000.0) (+ l (* w (- (* w l) l))) (/ 1.0 (/ (+ w 1.0) l))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 70000.0) {
		tmp = l + (w * ((w * l) - l));
	} else {
		tmp = 1.0 / ((w + 1.0) / 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 <= 70000.0d0) then
        tmp = l + (w * ((w * l) - l))
    else
        tmp = 1.0d0 / ((w + 1.0d0) / l)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 70000.0) {
		tmp = l + (w * ((w * l) - l));
	} else {
		tmp = 1.0 / ((w + 1.0) / l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 70000.0:
		tmp = l + (w * ((w * l) - l))
	else:
		tmp = 1.0 / ((w + 1.0) / l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 70000.0)
		tmp = Float64(l + Float64(w * Float64(Float64(w * l) - l)));
	else
		tmp = Float64(1.0 / Float64(Float64(w + 1.0) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 70000.0)
		tmp = l + (w * ((w * l) - l));
	else
		tmp = 1.0 / ((w + 1.0) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 70000.0], N[(l + N[(w * N[(N[(w * l), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(w + 1.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 7e4

   1. Initial program 98.6%

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

      1. add-sqr-sqrt 25.4%

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

      2. sqrt-unprod 79.6%

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

      3. sqr-neg 79.6%

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

      4. sqrt-unprod 54.2%

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

      5. add-sqr-sqrt 78.8%

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

      6. add-sqr-sqrt 78.8%

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

      7. sqrt-unprod 78.8%

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

      8. add-sqr-sqrt 54.2%

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

      9. sqrt-unprod 78.4%

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

      10. sqr-neg 78.4%

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

      11. sqrt-unprod 24.2%

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

      12. add-sqr-sqrt 63.3%

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

      13. pow1 63.3%

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

      14. exp-neg 63.3%

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

      15. inv-pow 63.3%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in w around inf 96.0%

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

      1. exp-neg 96.0%

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

      2. associate-*r/ 96.0%

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

      3. *-rgt-identity 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in w around 0 63.7%

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

      1. +-commutative 66.6%

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

   10. Simplified 63.7%

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

   11. Taylor expanded in w around 0 77.3%

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

   if 7e4 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 47.5%

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

      1. +-commutative 100.0%

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

   10. Simplified 47.5%

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

       1. clear-num 50.7%

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

       2. inv-pow 50.7%

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

   12. Applied egg-rr 50.7%

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

       1. unpow-1 50.7%

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

       2. +-commutative 50.7%

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

   14. Simplified 50.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{1 + w}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 70000:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.2% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.000235:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.000235) (- l (* w l)) (/ 1.0 (/ (+ w 1.0) l))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.000235) {
		tmp = l - (w * l);
	} else {
		tmp = 1.0 / ((w + 1.0) / 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 <= 0.000235d0) then
        tmp = l - (w * l)
    else
        tmp = 1.0d0 / ((w + 1.0d0) / l)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.000235) {
		tmp = l - (w * l);
	} else {
		tmp = 1.0 / ((w + 1.0) / l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.000235:
		tmp = l - (w * l)
	else:
		tmp = 1.0 / ((w + 1.0) / l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.000235)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(1.0 / Float64(Float64(w + 1.0) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.000235)
		tmp = l - (w * l);
	else
		tmp = 1.0 / ((w + 1.0) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.000235], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(w + 1.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.34999999999999993e-4

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 24.9%

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

      2. sqrt-unprod 80.2%

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

      3. sqr-neg 80.2%

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

      4. sqrt-unprod 55.3%

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

      5. add-sqr-sqrt 79.9%

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

      6. add-sqr-sqrt 79.9%

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

      7. sqrt-unprod 79.9%

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

      8. add-sqr-sqrt 55.3%

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

      9. sqrt-unprod 79.9%

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

      10. sqr-neg 79.9%

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

      11. sqrt-unprod 24.6%

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

      12. add-sqr-sqrt 64.6%

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

      13. pow1 64.6%

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

      14. exp-neg 64.6%

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

      15. inv-pow 64.6%

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

      16. pow-prod-up 97.4%

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

      17. metadata-eval 97.4%

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

      18. metadata-eval 97.4%

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

      19. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in w around inf 97.4%

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

      1. exp-neg 97.4%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in w around 0 64.9%

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

      1. +-commutative 66.4%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in w around 0 73.7%

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

       1. mul-1-neg 73.7%

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

       2. unsub-neg 73.7%

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

   13. Simplified 73.7%

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

   if 2.34999999999999993e-4 < w

   1. Initial program 96.7%

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

      1. add-sqr-sqrt 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. sqr-neg 96.7%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.1%

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

      6. add-sqr-sqrt 95.1%

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

      7. sqrt-unprod 95.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 95.2%

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

      17. metadata-eval 95.2%

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

      18. metadata-eval 95.2%

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

      19. metadata-eval 95.2%

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{1}} \]

   4. Applied egg-rr 95.2%

      \[\leadsto e^{-w} \cdot \color{blue}{\left(\ell \cdot 1\right)} \]

   5. Taylor expanded in w around inf 95.2%

      \[\leadsto \color{blue}{\ell \cdot e^{-w}} \]
   6. Step-by-step derivation

      1. exp-neg 95.2%

         \[\leadsto \ell \cdot \color{blue}{\frac{1}{e^{w}}} \]

      2. associate-*r/ 95.2%

         \[\leadsto \color{blue}{\frac{\ell \cdot 1}{e^{w}}} \]

      3. *-rgt-identity 95.2%

         \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   7. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 44.7%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 44.7%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]
   11. Step-by-step derivation

       1. clear-num 47.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{w + 1}{\ell}}} \]

       2. inv-pow 47.7%

          \[\leadsto \color{blue}{{\left(\frac{w + 1}{\ell}\right)}^{-1}} \]

   12. Applied egg-rr 47.7%

       \[\leadsto \color{blue}{{\left(\frac{w + 1}{\ell}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 47.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{w + 1}{\ell}}} \]

       2. +-commutative 47.7%

          \[\leadsto \frac{1}{\frac{\color{blue}{1 + w}}{\ell}} \]

   14. Simplified 47.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{1 + w}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.000235:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.6% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 10^{-11}:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1e-11) (- l (* w l)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1e-11) {
		tmp = l - (w * l);
	} else {
		tmp = l / (w + 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 <= 1d-11) then
        tmp = l - (w * l)
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 1e-11) {
		tmp = l - (w * l);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1e-11:
		tmp = l - (w * l)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1e-11)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1e-11)
		tmp = l - (w * l);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1e-11], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 9.99999999999999939e-12

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 24.6%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)} \]

      2. sqrt-unprod 80.2%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)} \]

      3. sqr-neg 80.2%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)} \]

      4. sqrt-unprod 55.6%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)} \]

      5. add-sqr-sqrt 80.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{-w}}\right)} \]

      6. add-sqr-sqrt 80.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}} \]

      7. sqrt-unprod 80.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}} \]

      8. add-sqr-sqrt 55.6%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)} \]

      9. sqrt-unprod 80.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)} \]

      10. sqr-neg 80.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)} \]

      11. sqrt-unprod 24.4%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)} \]

      12. add-sqr-sqrt 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)} \]

      13. pow1 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)} \]

      14. exp-neg 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)} \]

      15. inv-pow 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)} \]

      16. pow-prod-up 97.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)} \]

      17. metadata-eval 97.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)} \]

      18. metadata-eval 97.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{1}}\right)} \]

      19. metadata-eval 97.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{1}} \]

   4. Applied egg-rr 97.6%

      \[\leadsto e^{-w} \cdot \color{blue}{\left(\ell \cdot 1\right)} \]

   5. Taylor expanded in w around inf 97.6%

      \[\leadsto \color{blue}{\ell \cdot e^{-w}} \]
   6. Step-by-step derivation

      1. exp-neg 97.6%

         \[\leadsto \ell \cdot \color{blue}{\frac{1}{e^{w}}} \]

      2. associate-*r/ 97.6%

         \[\leadsto \color{blue}{\frac{\ell \cdot 1}{e^{w}}} \]

      3. *-rgt-identity 97.6%

         \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   7. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 64.9%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 66.3%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 64.9%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]

   11. Taylor expanded in w around 0 73.8%

       \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 73.8%

          \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

       2. unsub-neg 73.8%

          \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   13. Simplified 73.8%

       \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   if 9.99999999999999939e-12 < w

   1. Initial program 96.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 96.6%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)} \]

      2. sqrt-unprod 96.6%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)} \]

      3. sqr-neg 96.6%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)} \]

      5. add-sqr-sqrt 94.5%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{-w}}\right)} \]

      6. add-sqr-sqrt 94.5%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}} \]

      7. sqrt-unprod 94.5%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)} \]

      9. sqrt-unprod 1.1%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)} \]

      10. sqr-neg 1.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)} \]

      11. sqrt-unprod 1.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)} \]

      12. add-sqr-sqrt 1.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)} \]

      13. pow1 1.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)} \]

      14. exp-neg 1.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)} \]

      15. inv-pow 1.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)} \]

      16. pow-prod-up 94.7%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)} \]

      17. metadata-eval 94.7%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)} \]

      18. metadata-eval 94.7%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{1}}\right)} \]

      19. metadata-eval 94.7%

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{1}} \]

   4. Applied egg-rr 94.7%

      \[\leadsto e^{-w} \cdot \color{blue}{\left(\ell \cdot 1\right)} \]

   5. Taylor expanded in w around inf 94.7%

      \[\leadsto \color{blue}{\ell \cdot e^{-w}} \]
   6. Step-by-step derivation

      1. exp-neg 94.7%

         \[\leadsto \ell \cdot \color{blue}{\frac{1}{e^{w}}} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot 1}{e^{w}}} \]

      3. *-rgt-identity 94.7%

         \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 44.9%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 98.2%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 44.9%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 10^{-11}:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.6% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 1.0) (- l (* w l)) (/ l w)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.0) {
		tmp = l - (w * l);
	} else {
		tmp = 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 <= 1.0d0) then
        tmp = l - (w * l)
    else
        tmp = l / w
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 1.0) {
		tmp = l - (w * l);
	} else {
		tmp = l / w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.0:
		tmp = l - (w * l)
	else:
		tmp = l / w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.0)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(l / w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.0)
		tmp = l - (w * l);
	else
		tmp = l / w;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.0], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(l / w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 24.9%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)} \]

      2. sqrt-unprod 80.2%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)} \]

      3. sqr-neg 80.2%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)} \]

      4. sqrt-unprod 55.3%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)} \]

      5. add-sqr-sqrt 79.9%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{-w}}\right)} \]

      6. add-sqr-sqrt 79.9%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}} \]

      7. sqrt-unprod 79.9%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}} \]

      8. add-sqr-sqrt 55.3%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)} \]

      9. sqrt-unprod 79.9%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)} \]

      10. sqr-neg 79.9%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)} \]

      11. sqrt-unprod 24.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)} \]

      12. add-sqr-sqrt 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)} \]

      13. pow1 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)} \]

      14. exp-neg 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)} \]

      15. inv-pow 64.6%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)} \]

      16. pow-prod-up 97.4%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)} \]

      17. metadata-eval 97.4%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)} \]

      18. metadata-eval 97.4%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{1}}\right)} \]

      19. metadata-eval 97.4%

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{1}} \]

   4. Applied egg-rr 97.4%

      \[\leadsto e^{-w} \cdot \color{blue}{\left(\ell \cdot 1\right)} \]

   5. Taylor expanded in w around inf 97.4%

      \[\leadsto \color{blue}{\ell \cdot e^{-w}} \]
   6. Step-by-step derivation

      1. exp-neg 97.4%

         \[\leadsto \ell \cdot \color{blue}{\frac{1}{e^{w}}} \]

      2. associate-*r/ 97.4%

         \[\leadsto \color{blue}{\frac{\ell \cdot 1}{e^{w}}} \]

      3. *-rgt-identity 97.4%

         \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   7. Simplified 97.4%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 64.9%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 66.4%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 64.9%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]

   11. Taylor expanded in w around 0 73.7%

       \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 73.7%

          \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

       2. unsub-neg 73.7%

          \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   13. Simplified 73.7%

       \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   if 1 < w

   1. Initial program 96.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 96.7%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)} \]

      2. sqrt-unprod 96.7%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)} \]

      3. sqr-neg 96.7%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)} \]

      5. add-sqr-sqrt 95.1%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{-w}}\right)} \]

      6. add-sqr-sqrt 95.1%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}} \]

      7. sqrt-unprod 95.1%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)} \]

      9. sqrt-unprod 0.1%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)} \]

      10. sqr-neg 0.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)} \]

      11. sqrt-unprod 0.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)} \]

      12. add-sqr-sqrt 0.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)} \]

      13. pow1 0.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)} \]

      14. exp-neg 0.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)} \]

      15. inv-pow 0.1%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)} \]

      16. pow-prod-up 95.2%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)} \]

      17. metadata-eval 95.2%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)} \]

      18. metadata-eval 95.2%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{1}}\right)} \]

      19. metadata-eval 95.2%

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{1}} \]

   4. Applied egg-rr 95.2%

      \[\leadsto e^{-w} \cdot \color{blue}{\left(\ell \cdot 1\right)} \]

   5. Taylor expanded in w around inf 95.2%

      \[\leadsto \color{blue}{\ell \cdot e^{-w}} \]
   6. Step-by-step derivation

      1. exp-neg 95.2%

         \[\leadsto \ell \cdot \color{blue}{\frac{1}{e^{w}}} \]

      2. associate-*r/ 95.2%

         \[\leadsto \color{blue}{\frac{\ell \cdot 1}{e^{w}}} \]

      3. *-rgt-identity 95.2%

         \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   7. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 44.7%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 44.7%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]

   11. Taylor expanded in w around inf 44.7%

       \[\leadsto \color{blue}{\frac{\ell}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.3% accurate, 38.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 7800:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 7800.0) l (/ l w)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 7800.0) {
		tmp = l;
	} else {
		tmp = 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 <= 7800.0d0) then
        tmp = l
    else
        tmp = l / w
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 7800.0) {
		tmp = l;
	} else {
		tmp = l / w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 7800.0:
		tmp = l
	else:
		tmp = l / w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 7800.0)
		tmp = l;
	else
		tmp = Float64(l / w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 7800.0)
		tmp = l;
	else
		tmp = l / w;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 7800.0], l, N[(l / w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 7800

   1. Initial program 98.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 64.6%

      \[\leadsto \color{blue}{\ell} \]

   if 7800 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)} \]

      2. sqrt-unprod 100.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)} \]

      3. sqr-neg 100.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)} \]

      4. sqrt-unprod 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)} \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{\color{blue}{-w}}\right)} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)} \]

      9. sqrt-unprod 0.0%

         \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)} \]

      10. sqr-neg 0.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)} \]

      11. sqrt-unprod 0.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)} \]

      13. pow1 0.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)} \]

      14. exp-neg 0.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)} \]

      15. inv-pow 0.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)} \]

      16. pow-prod-up 100.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)} \]

      17. metadata-eval 100.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)} \]

      18. metadata-eval 100.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\left(\sqrt{\color{blue}{1}}\right)} \]

      19. metadata-eval 100.0%

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{1}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto e^{-w} \cdot \color{blue}{\left(\ell \cdot 1\right)} \]

   5. Taylor expanded in w around inf 100.0%

      \[\leadsto \color{blue}{\ell \cdot e^{-w}} \]
   6. Step-by-step derivation

      1. exp-neg 100.0%

         \[\leadsto \ell \cdot \color{blue}{\frac{1}{e^{w}}} \]

      2. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{\ell \cdot 1}{e^{w}}} \]

      3. *-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 47.5%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 47.5%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]

   11. Taylor expanded in w around inf 47.5%

       \[\leadsto \color{blue}{\frac{\ell}{w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 51.3% 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 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Taylor expanded in w around 0 51.4%

   \[\leadsto \color{blue}{\ell} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))