exp-w (used to crash)

Percentage Accurate: 99.4% → 99.5%

Time: 21.6s

Alternatives: 20

Speedup: 1.4×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (exp (- w))
   (/
    (pow l (exp w))
    (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, exp(w)) / (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 <= (-1.6d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** exp(w)) / (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 <= -1.6) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.6:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, math.exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.6)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / 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 <= -1.6)
		tmp = exp(-w);
	else
		tmp = (l ^ exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / 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 2 regimes

2. if w < -1.6000000000000001

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -1.6000000000000001 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}} \]
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 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.6%

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

   1. exp-neg 98.6%

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

   2. remove-double-neg 98.6%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. remove-double-neg 98.6%

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

3. Simplified 98.6%

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

Alternative 4: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.0)
   (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 <= -4.0) {
		tmp = 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 <= (-4.0d0)) then
        tmp = 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 <= -4.0) {
		tmp = 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 <= -4.0:
		tmp = 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 <= -4.0)
		tmp = exp(Float64(-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 <= -4.0)
		tmp = 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, -4.0], N[Exp[(-w)], $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 < -4

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -4 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

Alternative 5: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;w \leq -1.55:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666))))))))
   (if (<= w -1.55) (exp (- w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.55) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0)))))
    if (w <= (-1.55d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.55) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, t_0) / t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))))
	tmp = 0
	if w <= -1.55:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, t_0) / t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))))
	tmp = 0.0
	if (w <= -1.55)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	tmp = 0.0;
	if (w <= -1.55)
		tmp = exp(-w);
	else
		tmp = (l ^ t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.55], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.55000000000000004

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -1.55000000000000004 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in w around 0 99.1%

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

      1. *-commutative 99.3%

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

   10. Simplified 99.1%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(1 + w \cdot \left(1 + w \cdot 0.5\right)\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3)
   (exp (- w))
   (/
    (pow l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))
    (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * 0.5))))) / (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 <= (-1.3d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))) / (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 <= -1.3) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * 0.5))))) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.3:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, (1.0 + (w * (1.0 + (w * 0.5))))) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.3)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5))))) / 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 <= -1.3)
		tmp = exp(-w);
	else
		tmp = (l ^ (1.0 + (w * (1.0 + (w * 0.5))))) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 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 2 regimes

2. if w < -1.30000000000000004

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -1.30000000000000004 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in w around 0 99.0%

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

      1. *-commutative 99.2%

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

   10. Simplified 99.0%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot 0.5\right)\right)}}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (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 = exp(-w)
    else
        tmp = (l ** (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))) / (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 = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (w + 1.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[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] / 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. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -1 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.0%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0 99.0%

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

      1. *-commutative 99.3%

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

   10. Simplified 99.0%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}}}{w + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * 0.5))))) / (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 = exp(-w)
    else
        tmp = (l ** (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))) / (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 = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * 0.5))))) / (w + 1.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -1 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.0%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0 99.0%

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

      1. *-commutative 99.2%

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

   10. Simplified 99.0%

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

Alternative 9: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ Float64(w + 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 = exp(-w);
	else
		tmp = (l ^ (w + 1.0)) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(w + 1.0), $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. Step-by-step derivation

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.1%

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

      3. sqr-neg 47.1%

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

      4. sqrt-unprod 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. add-sqr-sqrt 47.1%

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

      7. sqrt-unprod 47.1%

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

      8. add-sqr-sqrt 47.1%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 95.7%

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

      17. metadata-eval 95.7%

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

      18. metadata-eval 95.7%

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

      19. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. add-exp-log 95.7%

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

      2. *-rgt-identity 95.7%

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

      3. log-div 95.7%

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

      4. add-log-exp 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in w around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   11. Simplified 99.7%

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

   if -1 < w

   1. Initial program 98.2%

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

      1. exp-neg 98.2%

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

      2. remove-double-neg 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-lft-identity 98.2%

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

      5. remove-double-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0 99.0%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 99.0%

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

   10. Simplified 98.8%

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

Alternative 10: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 620\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 620.0))) (exp (- w)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 620.0)) {
		tmp = exp(-w);
	} 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 <= (-0.7d0)) .or. (.not. (w <= 620.0d0))) then
        tmp = exp(-w)
    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 <= -0.7) || !(w <= 620.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 620.0):
		tmp = math.exp(-w)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 620.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 620.0)))
		tmp = exp(-w);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 620.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 620 < w

   1. Initial program 97.3%

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

      1. exp-neg 97.3%

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

      2. remove-double-neg 97.3%

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

      3. associate-*l/ 97.3%

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

      4. *-lft-identity 97.3%

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

      5. remove-double-neg 97.3%

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

   3. Simplified 97.3%

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

      1. add-sqr-sqrt 37.8%

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

      2. sqrt-unprod 65.9%

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

      3. sqr-neg 65.9%

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

      4. sqrt-unprod 28.0%

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

      5. add-sqr-sqrt 65.9%

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

      6. add-sqr-sqrt 65.9%

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

      7. sqrt-unprod 65.9%

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

      8. add-sqr-sqrt 28.0%

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

      9. sqrt-unprod 28.0%

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

      10. sqr-neg 28.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 94.8%

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

      17. metadata-eval 94.8%

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

      18. metadata-eval 94.8%

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

      19. metadata-eval 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. add-exp-log 94.8%

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

      2. *-rgt-identity 94.8%

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

      3. log-div 94.8%

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

      4. add-log-exp 94.8%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in w around inf 97.2%

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

       1. neg-mul-1 97.2%

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

   11. Simplified 97.2%

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

   if -0.69999999999999996 < w < 620

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 49.4%

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

      2. sqrt-unprod 98.6%

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

      3. sqr-neg 98.6%

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

      4. sqrt-unprod 49.2%

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

      5. add-sqr-sqrt 97.1%

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

      6. add-sqr-sqrt 97.1%

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

      7. sqrt-unprod 97.1%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-unprod 97.1%

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

      10. sqr-neg 97.1%

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

      11. sqrt-unprod 47.9%

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

      12. add-sqr-sqrt 97.1%

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

      13. pow1 97.1%

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

      14. exp-neg 97.1%

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

      15. inv-pow 97.1%

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

      16. pow-prod-up 97.1%

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

      17. metadata-eval 97.1%

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

      18. metadata-eval 97.1%

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

      19. metadata-eval 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in l around 0 97.1%

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

   8. Taylor expanded in w around 0 97.1%

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

      1. +-commutative 98.7%

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

   10. Simplified 97.1%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 620\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 98.6%

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

   1. exp-neg 98.6%

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

   2. remove-double-neg 98.6%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. remove-double-neg 98.6%

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

3. Simplified 98.6%

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

   1. add-sqr-sqrt 44.4%

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

   2. sqrt-unprod 84.4%

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

   3. sqr-neg 84.4%

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

   4. sqrt-unprod 40.0%

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

   5. add-sqr-sqrt 83.6%

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

   6. add-sqr-sqrt 83.6%

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

   7. sqrt-unprod 83.6%

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

   8. add-sqr-sqrt 40.0%

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

   9. sqrt-unprod 67.2%

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

   10. sqr-neg 67.2%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 55.1%

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

   13. pow1 55.1%

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

   14. exp-neg 55.1%

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

   15. inv-pow 55.1%

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

   16. pow-prod-up 96.1%

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

   17. metadata-eval 96.1%

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

   18. metadata-eval 96.1%

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

   19. metadata-eval 96.1%

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

6. Applied egg-rr 96.1%

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

7. Taylor expanded in l around 0 96.1%

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

Alternative 12: 90.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.12)
   (+ l (* l (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.12) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.12d0) then
        tmp = l + (l * (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.12) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.12:
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.12)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.12)
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.12], N[(l + N[(l * N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.12

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 33.9%

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

      2. sqrt-unprod 82.5%

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

      3. sqr-neg 82.5%

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

      4. sqrt-unprod 48.6%

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

      5. add-sqr-sqrt 81.5%

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

      6. add-sqr-sqrt 81.5%

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

      7. sqrt-unprod 81.5%

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

      8. add-sqr-sqrt 48.6%

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

      9. sqrt-unprod 81.5%

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

      10. sqr-neg 81.5%

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

      11. sqrt-unprod 32.9%

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

      12. add-sqr-sqrt 66.8%

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

      13. pow1 66.8%

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

      14. exp-neg 66.8%

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

      15. inv-pow 66.8%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in w around 0 87.7%

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

   8. Taylor expanded in l around 0 89.4%

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

   if 0.12 < w

   1. Initial program 93.3%

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

      1. exp-neg 93.3%

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

      2. remove-double-neg 93.3%

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

      3. associate-*l/ 93.3%

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

      4. *-lft-identity 93.3%

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

      5. remove-double-neg 93.3%

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

   3. Simplified 93.3%

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

      1. add-sqr-sqrt 93.3%

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

      2. sqrt-unprod 93.3%

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

      3. sqr-neg 93.3%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 93.4%

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

      6. add-sqr-sqrt 93.4%

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

      7. sqrt-unprod 93.4%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 93.4%

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

      17. metadata-eval 93.4%

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

      18. metadata-eval 93.4%

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

      19. metadata-eval 93.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in w around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. unsub-neg 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in w around inf 3.2%

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

       1. neg-mul-1 3.2%

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

       2. distribute-lft-neg-in 3.2%

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

       3. *-commutative 3.2%

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

   12. Simplified 3.2%

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

       1. neg-mul-1 3.2%

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

       2. expm1-log1p-u 3.2%

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

       3. expm1-undefine 89.0%

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

       4. neg-mul-1 89.0%

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

   14. Applied egg-rr 89.0%

       \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. sub-neg 89.0%

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

       2. metadata-eval 89.0%

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

       3. +-commutative 89.0%

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

       4. log1p-undefine 89.0%

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

       5. rem-exp-log 89.0%

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

       6. unsub-neg 89.0%

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

   16. Simplified 89.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.7% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.25:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.25)
   (+ l (* l (* w (+ (* w (* w -0.16666666666666666)) -1.0))))
   (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.25) {
		tmp = l + (l * (w * ((w * (w * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.25d0) then
        tmp = l + (l * (w * ((w * (w * (-0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.25) {
		tmp = l + (l * (w * ((w * (w * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.25:
		tmp = l + (l * (w * ((w * (w * -0.16666666666666666)) + -1.0)))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.25)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(w * -0.16666666666666666)) + -1.0))));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.25)
		tmp = l + (l * (w * ((w * (w * -0.16666666666666666)) + -1.0)));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.25], N[(l + N[(l * N[(w * N[(N[(w * N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.25

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 33.9%

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

      2. sqrt-unprod 82.5%

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

      3. sqr-neg 82.5%

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

      4. sqrt-unprod 48.6%

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

      5. add-sqr-sqrt 81.5%

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

      6. add-sqr-sqrt 81.5%

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

      7. sqrt-unprod 81.5%

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

      8. add-sqr-sqrt 48.6%

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

      9. sqrt-unprod 81.5%

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

      10. sqr-neg 81.5%

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

      11. sqrt-unprod 32.9%

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

      12. add-sqr-sqrt 66.8%

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

      13. pow1 66.8%

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

      14. exp-neg 66.8%

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

      15. inv-pow 66.8%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in w around 0 87.7%

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

   8. Taylor expanded in l around 0 89.4%

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

   9. Taylor expanded in w around inf 89.4%

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

       1. *-commutative 89.4%

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

   11. Simplified 89.4%

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

   if 0.25 < w

   1. Initial program 93.3%

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

      1. exp-neg 93.3%

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

      2. remove-double-neg 93.3%

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

      3. associate-*l/ 93.3%

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

      4. *-lft-identity 93.3%

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

      5. remove-double-neg 93.3%

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

   3. Simplified 93.3%

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

      1. add-sqr-sqrt 93.3%

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

      2. sqrt-unprod 93.3%

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

      3. sqr-neg 93.3%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 93.4%

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

      6. add-sqr-sqrt 93.4%

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

      7. sqrt-unprod 93.4%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 93.4%

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

      17. metadata-eval 93.4%

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

      18. metadata-eval 93.4%

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

      19. metadata-eval 93.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in w around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. unsub-neg 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in w around inf 3.2%

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

       1. neg-mul-1 3.2%

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

       2. distribute-lft-neg-in 3.2%

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

       3. *-commutative 3.2%

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

   12. Simplified 3.2%

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

       1. neg-mul-1 3.2%

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

       2. expm1-log1p-u 3.2%

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

       3. expm1-undefine 89.0%

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

       4. neg-mul-1 89.0%

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

   14. Applied egg-rr 89.0%

       \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. sub-neg 89.0%

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

       2. metadata-eval 89.0%

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

       3. +-commutative 89.0%

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

       4. log1p-undefine 89.0%

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

       5. rem-exp-log 89.0%

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

       6. unsub-neg 89.0%

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

   16. Simplified 89.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.25:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.23:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.23)
   (* l (+ 1.0 (* w (+ (* w 0.5) -1.0))))
   (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.23) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.23d0) then
        tmp = l * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.23) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.23:
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.23)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.23)
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.23], N[(l * N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.23000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0 82.3%

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

      1. add-sqr-sqrt 33.9%

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

      2. sqrt-unprod 82.5%

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

      3. sqr-neg 82.5%

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

      4. sqrt-unprod 48.6%

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

      5. add-sqr-sqrt 81.5%

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

      6. add-sqr-sqrt 81.5%

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

      7. sqrt-unprod 81.5%

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

      8. add-sqr-sqrt 48.6%

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

      9. sqrt-unprod 81.5%

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

      10. sqr-neg 81.5%

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

      11. sqrt-unprod 32.9%

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

      12. add-sqr-sqrt 66.8%

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

      13. pow1 66.8%

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

      14. exp-neg 66.8%

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

      15. inv-pow 66.8%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   5. Applied egg-rr 87.3%

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

   if 0.23000000000000001 < w

   1. Initial program 93.3%

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

      1. exp-neg 93.3%

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

      2. remove-double-neg 93.3%

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

      3. associate-*l/ 93.3%

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

      4. *-lft-identity 93.3%

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

      5. remove-double-neg 93.3%

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

   3. Simplified 93.3%

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

      1. add-sqr-sqrt 93.3%

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

      2. sqrt-unprod 93.3%

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

      3. sqr-neg 93.3%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 93.4%

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

      6. add-sqr-sqrt 93.4%

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

      7. sqrt-unprod 93.4%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 93.4%

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

      17. metadata-eval 93.4%

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

      18. metadata-eval 93.4%

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

      19. metadata-eval 93.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in w around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. unsub-neg 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in w around inf 3.2%

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

       1. neg-mul-1 3.2%

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

       2. distribute-lft-neg-in 3.2%

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

       3. *-commutative 3.2%

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

   12. Simplified 3.2%

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

       1. neg-mul-1 3.2%

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

       2. expm1-log1p-u 3.2%

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

       3. expm1-undefine 89.0%

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

       4. neg-mul-1 89.0%

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

   14. Applied egg-rr 89.0%

       \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. sub-neg 89.0%

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

       2. metadata-eval 89.0%

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

       3. +-commutative 89.0%

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

       4. log1p-undefine 89.0%

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

       5. rem-exp-log 89.0%

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

       6. unsub-neg 89.0%

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

   16. Simplified 89.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.23:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.18:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.18)
   (+ l (* l (* w (+ (* w 0.5) -1.0))))
   (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.18) {
		tmp = l + (l * (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.18d0) then
        tmp = l + (l * (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.18) {
		tmp = l + (l * (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.18:
		tmp = l + (l * (w * ((w * 0.5) + -1.0)))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.18)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.18)
		tmp = l + (l * (w * ((w * 0.5) + -1.0)));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.18], N[(l + N[(l * N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.17999999999999999

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 33.9%

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

      2. sqrt-unprod 82.5%

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

      3. sqr-neg 82.5%

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

      4. sqrt-unprod 48.6%

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

      5. add-sqr-sqrt 81.5%

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

      6. add-sqr-sqrt 81.5%

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

      7. sqrt-unprod 81.5%

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

      8. add-sqr-sqrt 48.6%

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

      9. sqrt-unprod 81.5%

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

      10. sqr-neg 81.5%

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

      11. sqrt-unprod 32.9%

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

      12. add-sqr-sqrt 66.8%

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

      13. pow1 66.8%

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

      14. exp-neg 66.8%

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

      15. inv-pow 66.8%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in w around 0 87.7%

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

   8. Taylor expanded in l around 0 89.4%

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

   9. Taylor expanded in w around 0 87.3%

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

       1. *-commutative 87.3%

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

   11. Simplified 87.3%

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

   if 0.17999999999999999 < w

   1. Initial program 93.3%

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

      1. exp-neg 93.3%

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

      2. remove-double-neg 93.3%

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

      3. associate-*l/ 93.3%

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

      4. *-lft-identity 93.3%

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

      5. remove-double-neg 93.3%

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

   3. Simplified 93.3%

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

      1. add-sqr-sqrt 93.3%

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

      2. sqrt-unprod 93.3%

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

      3. sqr-neg 93.3%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 93.4%

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

      6. add-sqr-sqrt 93.4%

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

      7. sqrt-unprod 93.4%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 93.4%

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

      17. metadata-eval 93.4%

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

      18. metadata-eval 93.4%

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

      19. metadata-eval 93.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in w around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. unsub-neg 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in w around inf 3.2%

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

       1. neg-mul-1 3.2%

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

       2. distribute-lft-neg-in 3.2%

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

       3. *-commutative 3.2%

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

   12. Simplified 3.2%

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

       1. neg-mul-1 3.2%

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

       2. expm1-log1p-u 3.2%

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

       3. expm1-undefine 89.0%

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

       4. neg-mul-1 89.0%

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

   14. Applied egg-rr 89.0%

       \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. sub-neg 89.0%

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

       2. metadata-eval 89.0%

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

       3. +-commutative 89.0%

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

       4. log1p-undefine 89.0%

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

       5. rem-exp-log 89.0%

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

       6. unsub-neg 89.0%

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

   16. Simplified 89.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.18:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.2% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.17) (* l (- 1.0 w)) (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l * (1.0 - w);
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.17d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l * (1.0 - w);
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.17:
		tmp = l * (1.0 - w)
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.17)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.17)
		tmp = l * (1.0 - w);
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.17], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.170000000000000012

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 33.9%

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

      2. sqrt-unprod 82.5%

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

      3. sqr-neg 82.5%

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

      4. sqrt-unprod 48.6%

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

      5. add-sqr-sqrt 81.5%

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

      6. add-sqr-sqrt 81.5%

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

      7. sqrt-unprod 81.5%

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

      8. add-sqr-sqrt 48.6%

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

      9. sqrt-unprod 81.5%

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

      10. sqr-neg 81.5%

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

      11. sqrt-unprod 32.9%

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

      12. add-sqr-sqrt 66.8%

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

      13. pow1 66.8%

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

      14. exp-neg 66.8%

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

      15. inv-pow 66.8%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in l around 0 96.7%

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

   8. Taylor expanded in w around 0 74.9%

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

      1. neg-mul-1 74.9%

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

      2. sub-neg 74.9%

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

      3. *-rgt-identity 74.9%

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

      4. distribute-lft-out-- 74.9%

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

   10. Simplified 74.9%

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

   if 0.170000000000000012 < w

   1. Initial program 93.3%

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

      1. exp-neg 93.3%

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

      2. remove-double-neg 93.3%

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

      3. associate-*l/ 93.3%

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

      4. *-lft-identity 93.3%

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

      5. remove-double-neg 93.3%

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

   3. Simplified 93.3%

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

      1. add-sqr-sqrt 93.3%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

      2. sqrt-unprod 93.3%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

      3. sqr-neg 93.3%

         \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

      5. add-sqr-sqrt 93.4%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

      6. add-sqr-sqrt 93.4%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

      7. sqrt-unprod 93.4%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      9. sqrt-unprod 0.0%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      10. sqr-neg 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

      13. pow1 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

      14. exp-neg 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

      15. inv-pow 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

      16. pow-prod-up 93.4%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

      17. metadata-eval 93.4%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

      18. metadata-eval 93.4%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

      19. metadata-eval 93.4%

          \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

   6. Applied egg-rr 93.4%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

   7. Taylor expanded in w around 0 3.2%

      \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 3.2%

         \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

      2. unsub-neg 3.2%

         \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   9. Simplified 3.2%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   10. Taylor expanded in w around inf 3.2%

       \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot w\right)} \]
   11. Step-by-step derivation

       1. neg-mul-1 3.2%

          \[\leadsto \color{blue}{-\ell \cdot w} \]

       2. distribute-lft-neg-in 3.2%

          \[\leadsto \color{blue}{\left(-\ell\right) \cdot w} \]

       3. *-commutative 3.2%

          \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   12. Simplified 3.2%

       \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]
   13. Step-by-step derivation

       1. neg-mul-1 3.2%

          \[\leadsto w \cdot \color{blue}{\left(-1 \cdot \ell\right)} \]

       2. expm1-log1p-u 3.2%

          \[\leadsto w \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \ell\right)\right)} \]

       3. expm1-undefine 89.0%

          \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \ell\right)} - 1\right)} \]

       4. neg-mul-1 89.0%

          \[\leadsto w \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{-\ell}\right)} - 1\right) \]

   14. Applied egg-rr 89.0%

       \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. sub-neg 89.0%

          \[\leadsto w \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} + \left(-1\right)\right)} \]

       2. metadata-eval 89.0%

          \[\leadsto w \cdot \left(e^{\mathsf{log1p}\left(-\ell\right)} + \color{blue}{-1}\right) \]

       3. +-commutative 89.0%

          \[\leadsto w \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(-\ell\right)}\right)} \]

       4. log1p-undefine 89.0%

          \[\leadsto w \cdot \left(-1 + e^{\color{blue}{\log \left(1 + \left(-\ell\right)\right)}}\right) \]

       5. rem-exp-log 89.0%

          \[\leadsto w \cdot \left(-1 + \color{blue}{\left(1 + \left(-\ell\right)\right)}\right) \]

       6. unsub-neg 89.0%

          \[\leadsto w \cdot \left(-1 + \color{blue}{\left(1 - \ell\right)}\right) \]

   16. Simplified 89.0%

       \[\leadsto w \cdot \color{blue}{\left(-1 + \left(1 - \ell\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.102:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.102) (* l (- 1.0 w)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.102) {
		tmp = l * (1.0 - w);
	} 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 <= (-0.102d0)) then
        tmp = l * (1.0d0 - w)
    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 <= -0.102) {
		tmp = l * (1.0 - w);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.102:
		tmp = l * (1.0 - w)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.102)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.102)
		tmp = l * (1.0 - w);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.102], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.101999999999999993

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

      5. remove-double-neg 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

      2. sqrt-unprod 47.1%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

      3. sqr-neg 47.1%

         \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

      4. sqrt-unprod 47.1%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

      5. add-sqr-sqrt 47.1%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

      6. add-sqr-sqrt 47.1%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

      7. sqrt-unprod 47.1%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

      8. add-sqr-sqrt 47.1%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      9. sqrt-unprod 47.1%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      10. sqr-neg 47.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      12. add-sqr-sqrt 0.2%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

      13. pow1 0.2%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

      14. exp-neg 0.2%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

      15. inv-pow 0.2%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

      16. pow-prod-up 95.7%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

      17. metadata-eval 95.7%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

      18. metadata-eval 95.7%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

      19. metadata-eval 95.7%

          \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

   6. Applied egg-rr 95.7%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

   7. Taylor expanded in l around 0 95.7%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 26.1%

      \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   9. Step-by-step derivation

      1. neg-mul-1 26.1%

         \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

      2. sub-neg 26.1%

         \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

      3. *-rgt-identity 26.1%

         \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

      4. distribute-lft-out-- 26.1%

         \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]

   10. Simplified 26.1%

       \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]

   if -0.101999999999999993 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 98.2%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. remove-double-neg 98.2%

         \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      3. associate-*l/ 98.2%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

      4. *-lft-identity 98.2%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

      5. remove-double-neg 98.2%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 59.8%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

      2. sqrt-unprod 97.4%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

      3. sqr-neg 97.4%

         \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

      4. sqrt-unprod 37.6%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

      5. add-sqr-sqrt 96.2%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

      6. add-sqr-sqrt 96.2%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

      7. sqrt-unprod 96.2%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

      8. add-sqr-sqrt 37.6%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      9. sqrt-unprod 74.1%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      10. sqr-neg 74.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      11. sqrt-unprod 36.5%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      12. add-sqr-sqrt 74.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

      13. pow1 74.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

      14. exp-neg 74.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

      15. inv-pow 74.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

      16. pow-prod-up 96.3%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

      17. metadata-eval 96.3%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

      18. metadata-eval 96.3%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

      19. metadata-eval 96.3%

          \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

   6. Applied egg-rr 96.3%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

   7. Taylor expanded in l around 0 96.3%

      \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

   8. Taylor expanded in w around 0 87.2%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   9. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]

   10. Simplified 87.2%

       \[\leadsto \frac{\ell}{\color{blue}{w + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 64.4% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -250:\\ \;\;\;\;\left(-w\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -250.0) (* (- w) l) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -250.0) {
		tmp = -w * l;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-250.0d0)) then
        tmp = -w * l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -250.0) {
		tmp = -w * l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -250.0:
		tmp = -w * l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -250.0)
		tmp = Float64(Float64(-w) * l);
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -250.0)
		tmp = -w * l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -250.0], N[((-w) * l), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -250

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

      5. remove-double-neg 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

      2. sqrt-unprod 48.5%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

      3. sqr-neg 48.5%

         \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

      4. sqrt-unprod 48.5%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

      5. add-sqr-sqrt 48.5%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

      6. add-sqr-sqrt 48.5%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

      7. sqrt-unprod 48.5%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

      8. add-sqr-sqrt 48.5%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      9. sqrt-unprod 48.5%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      10. sqr-neg 48.5%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      12. add-sqr-sqrt 0.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

      13. pow1 0.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

      14. exp-neg 0.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

      15. inv-pow 0.1%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

      16. pow-prod-up 98.5%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

      17. metadata-eval 98.5%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

      18. metadata-eval 98.5%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

      19. metadata-eval 98.5%

          \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

   6. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

   7. Taylor expanded in w around 0 26.8%

      \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 26.8%

         \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

      2. unsub-neg 26.8%

         \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   9. Simplified 26.8%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   10. Taylor expanded in w around inf 26.8%

       \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot w\right)} \]
   11. Step-by-step derivation

       1. neg-mul-1 26.8%

          \[\leadsto \color{blue}{-\ell \cdot w} \]

       2. distribute-lft-neg-in 26.8%

          \[\leadsto \color{blue}{\left(-\ell\right) \cdot w} \]

       3. *-commutative 26.8%

          \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   12. Simplified 26.8%

       \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   if -250 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 74.7%

      \[\leadsto \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -250:\\ \;\;\;\;\left(-w\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.1% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 - w\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (- 1.0 w)))

C

double code(double w, double l) {
	return l * (1.0 - w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 - w)
end function

Java

public static double code(double w, double l) {
	return l * (1.0 - w);
}

Python

def code(w, l):
	return l * (1.0 - w)

Julia

function code(w, l)
	return Float64(l * Float64(1.0 - w))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 - w);
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 98.6%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 98.6%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 98.6%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 98.6%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 44.4%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.4%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.4%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 40.0%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 83.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 83.6%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 83.6%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 40.0%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 67.2%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 67.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 27.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 55.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 55.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 55.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 55.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 96.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 96.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 96.1%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 96.1%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 96.1%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

7. Taylor expanded in l around 0 96.1%

   \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

8. Taylor expanded in w around 0 62.3%

   \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 62.3%

      \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

   2. sub-neg 62.3%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   3. *-rgt-identity 62.3%

      \[\leadsto \color{blue}{\ell \cdot 1} - \ell \cdot w \]

   4. distribute-lft-out-- 62.3%

      \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]

10. Simplified 62.3%

    \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]
11. Add Preprocessing

Alternative 20: 57.2% 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 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Taylor expanded in w around 0 57.0%

   \[\leadsto \color{blue}{\ell} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))