exp-w (used to crash)

Percentage Accurate: 99.4% → 99.3%

Time: 14.4s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{e^{w}}\\ \frac{{\left({\ell}^{t\_0}\right)}^{t\_0}}{e^{w}} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (sqrt (exp w)))) (/ (pow (pow l t_0) t_0) (exp w))))

C

double code(double w, double l) {
	double t_0 = sqrt(exp(w));
	return pow(pow(l, t_0), t_0) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = sqrt(exp(w))
    code = ((l ** t_0) ** t_0) / exp(w)
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.sqrt(Math.exp(w));
	return Math.pow(Math.pow(l, t_0), t_0) / Math.exp(w);
}

Python

def code(w, l):
	t_0 = math.sqrt(math.exp(w))
	return math.pow(math.pow(l, t_0), t_0) / math.exp(w)

Julia

function code(w, l)
	t_0 = sqrt(exp(w))
	return Float64(((l ^ t_0) ^ t_0) / exp(w))
end

MATLAB

function tmp = code(w, l)
	t_0 = sqrt(exp(w));
	tmp = ((l ^ t_0) ^ t_0) / exp(w);
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[l, t$95$0], $MachinePrecision], t$95$0], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

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. Taylor expanded in l around inf 94.8%

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

   1. mul-1-neg 94.8%

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

   2. *-commutative 94.8%

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

   3. distribute-lft-neg-in 94.8%

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

   4. log-rec 94.8%

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

   5. remove-double-div 94.8%

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

7. Simplified 94.8%

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

   1. pow-to-exp 99.7%

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

   2. add-sqr-sqrt 99.6%

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

   3. pow-unpow 99.7%

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

9. Applied egg-rr 99.7%

   \[\leadsto \frac{\color{blue}{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}}{e^{w}} \]
10. 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 99.7%

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

Alternative 4: 97.6% accurate, 2.9× speedup

Math

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

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(Float64(-w)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 44.7%

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

   2. sqrt-unprod 82.7%

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

   3. sqr-neg 82.7%

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

   4. sqrt-unprod 38.0%

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

   5. add-sqr-sqrt 81.8%

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

   6. add-sqr-sqrt 81.8%

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

   7. sqrt-unprod 81.8%

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

   8. add-sqr-sqrt 38.0%

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

   9. sqrt-unprod 66.6%

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

   10. sqr-neg 66.6%

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

   11. sqrt-unprod 28.6%

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

   12. add-sqr-sqrt 56.9%

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

   13. pow1 56.9%

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

   14. exp-neg 56.9%

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

   15. inv-pow 56.9%

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

   16. pow-prod-up 97.9%

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

   17. metadata-eval 97.9%

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

   18. metadata-eval 97.9%

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

   19. metadata-eval 97.9%

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

4. Applied egg-rr 97.9%

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

5. Taylor expanded in w around inf 97.9%

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

Alternative 5: 97.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 44.7%

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

   2. sqrt-unprod 82.7%

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

   3. sqr-neg 82.7%

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

   4. sqrt-unprod 38.0%

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

   5. add-sqr-sqrt 81.8%

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

   6. add-sqr-sqrt 81.8%

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

   7. sqrt-unprod 81.8%

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

   8. add-sqr-sqrt 38.0%

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

   9. sqrt-unprod 66.6%

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

   10. sqr-neg 66.6%

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

   11. sqrt-unprod 28.6%

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

   12. add-sqr-sqrt 56.9%

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

   13. pow1 56.9%

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

   14. exp-neg 56.9%

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

   15. inv-pow 56.9%

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

   16. pow-prod-up 97.9%

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

   17. metadata-eval 97.9%

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

   18. metadata-eval 97.9%

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

   19. metadata-eval 97.9%

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

4. Applied egg-rr 97.9%

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

5. Taylor expanded in w around inf 97.9%

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

   1. exp-neg 97.9%

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

   2. associate-*r/ 97.9%

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

   3. *-rgt-identity 97.9%

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

   4. rem-exp-log 93.0%

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

   5. rem-exp-log 97.9%

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

7. Simplified 97.9%

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

Alternative 6: 84.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.8e-6)
   (* l (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.8e-6) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} 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 <= 1.8d-6) then
        tmp = l * (1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= 1.8e-6) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.8e-6:
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.8e-6)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.8e-6)
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.8e-6], N[(l * N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.79999999999999992e-6

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 79.5%

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

      3. sqr-neg 79.5%

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

      4. sqrt-unprod 45.2%

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

      5. add-sqr-sqrt 79.0%

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

      6. add-sqr-sqrt 79.0%

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

      7. sqrt-unprod 79.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. sqrt-unprod 79.0%

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

      10. sqr-neg 79.0%

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

      11. sqrt-unprod 33.8%

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

      12. add-sqr-sqrt 67.4%

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

      13. pow1 67.4%

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

      14. exp-neg 67.4%

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

      15. inv-pow 67.4%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 90.5%

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

   if 1.79999999999999992e-6 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. sqr-neg 99.5%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 96.6%

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

      6. add-sqr-sqrt 96.6%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.6%

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

      10. sqr-neg 1.6%

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

      11. sqrt-unprod 1.6%

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

      12. add-sqr-sqrt 1.6%

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

      13. pow1 1.6%

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

      14. exp-neg 1.6%

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

      15. inv-pow 1.6%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.9e-6)
   (+ l (* l (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.9e-6) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} 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 <= 1.9d-6) then
        tmp = l + (l * (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= 1.9e-6) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.9e-6:
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.9e-6)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.9e-6)
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.9e-6], 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[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.9e-6

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 79.5%

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

      3. sqr-neg 79.5%

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

      4. sqrt-unprod 45.2%

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

      5. add-sqr-sqrt 79.0%

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

      6. add-sqr-sqrt 79.0%

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

      7. sqrt-unprod 79.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. sqrt-unprod 79.0%

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

      10. sqr-neg 79.0%

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

      11. sqrt-unprod 33.8%

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

      12. add-sqr-sqrt 67.4%

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

      13. pow1 67.4%

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

      14. exp-neg 67.4%

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

      15. inv-pow 67.4%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 88.0%

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

   6. Taylor expanded in l around 0 90.5%

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

   if 1.9e-6 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. sqr-neg 99.5%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 96.6%

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

      6. add-sqr-sqrt 96.6%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.6%

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

      10. sqr-neg 1.6%

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

      11. sqrt-unprod 1.6%

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

      12. add-sqr-sqrt 1.6%

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

      13. pow1 1.6%

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

      14. exp-neg 1.6%

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

      15. inv-pow 1.6%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.75e-6) (* l (+ 1.0 (* w (+ (* w 0.5) -1.0)))) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.75e-6) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} 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 <= 1.75d-6) then
        tmp = l * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    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 <= 1.75e-6) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.75e-6:
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.75e-6)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.75e-6)
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.75e-6], N[(l * N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.74999999999999997e-6

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 79.5%

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

      3. sqr-neg 79.5%

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

      4. sqrt-unprod 45.2%

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

      5. add-sqr-sqrt 79.0%

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

      6. add-sqr-sqrt 79.0%

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

      7. sqrt-unprod 79.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. sqrt-unprod 79.0%

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

      10. sqr-neg 79.0%

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

      11. sqrt-unprod 33.8%

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

      12. add-sqr-sqrt 67.4%

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

      13. pow1 67.4%

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

      14. exp-neg 67.4%

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

      15. inv-pow 67.4%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 86.1%

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

   if 1.74999999999999997e-6 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. sqr-neg 99.5%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 96.6%

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

      6. add-sqr-sqrt 96.6%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.6%

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

      10. sqr-neg 1.6%

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

      11. sqrt-unprod 1.6%

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

      12. add-sqr-sqrt 1.6%

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

      13. pow1 1.6%

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

      14. exp-neg 1.6%

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

      15. inv-pow 1.6%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.8e-6) (+ l (* w (* l (+ (* w 0.5) -1.0)))) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.8e-6) {
		tmp = l + (w * (l * ((w * 0.5) + -1.0)));
	} 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 <= 1.8d-6) then
        tmp = l + (w * (l * ((w * 0.5d0) + (-1.0d0))))
    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 <= 1.8e-6) {
		tmp = l + (w * (l * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.8e-6:
		tmp = l + (w * (l * ((w * 0.5) + -1.0)))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.8e-6)
		tmp = Float64(l + Float64(w * Float64(l * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 1.8e-6)
		tmp = l + (w * (l * ((w * 0.5) + -1.0)));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.8e-6], N[(l + N[(w * N[(l * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.79999999999999992e-6

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 79.5%

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

      3. sqr-neg 79.5%

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

      4. sqrt-unprod 45.2%

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

      5. add-sqr-sqrt 79.0%

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

      6. add-sqr-sqrt 79.0%

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

      7. sqrt-unprod 79.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. sqrt-unprod 79.0%

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

      10. sqr-neg 79.0%

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

      11. sqrt-unprod 33.8%

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

      12. add-sqr-sqrt 67.4%

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

      13. pow1 67.4%

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

      14. exp-neg 67.4%

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

      15. inv-pow 67.4%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 88.0%

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

   6. Taylor expanded in w around 0 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*r* 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-*r* 82.2%

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

      5. distribute-lft-out 82.2%

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

      6. *-commutative 82.2%

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

   8. Simplified 82.2%

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

   if 1.79999999999999992e-6 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. sqr-neg 99.5%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 96.6%

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

      6. add-sqr-sqrt 96.6%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.6%

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

      10. sqr-neg 1.6%

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

      11. sqrt-unprod 1.6%

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

      12. add-sqr-sqrt 1.6%

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

      13. pow1 1.6%

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

      14. exp-neg 1.6%

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

      15. inv-pow 1.6%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.9% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\ell \cdot \left(-1 + \left(2 - w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.8e-6) (* l (+ -1.0 (- 2.0 w))) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.8e-6) {
		tmp = l * (-1.0 + (2.0 - 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 <= 1.8d-6) then
        tmp = l * ((-1.0d0) + (2.0d0 - 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 <= 1.8e-6) {
		tmp = l * (-1.0 + (2.0 - w));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.8e-6:
		tmp = l * (-1.0 + (2.0 - w))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.8e-6)
		tmp = Float64(l * Float64(-1.0 + Float64(2.0 - 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 <= 1.8e-6)
		tmp = l * (-1.0 + (2.0 - w));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.8e-6], N[(l * N[(-1.0 + N[(2.0 - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.79999999999999992e-6

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 79.5%

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

      3. sqr-neg 79.5%

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

      4. sqrt-unprod 45.2%

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

      5. add-sqr-sqrt 79.0%

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

      6. add-sqr-sqrt 79.0%

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

      7. sqrt-unprod 79.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. sqrt-unprod 79.0%

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

      10. sqr-neg 79.0%

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

      11. sqrt-unprod 33.8%

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

      12. add-sqr-sqrt 67.4%

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

      13. pow1 67.4%

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

      14. exp-neg 67.4%

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

      15. inv-pow 67.4%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 88.0%

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

   6. Taylor expanded in w around 0 73.6%

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

      1. *-rgt-identity 73.6%

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

      2. neg-mul-1 73.6%

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

      3. distribute-rgt-neg-in 73.6%

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

      4. distribute-lft-out 73.6%

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

      5. sub-neg 73.6%

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

   8. Simplified 73.6%

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

      1. expm1-log1p-u 73.6%

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

   10. Applied egg-rr 73.6%

       \[\leadsto \ell \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - w\right)\right)} \]
   11. Step-by-step derivation

       1. expm1-undefine 73.6%

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

       2. sub-neg 73.6%

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

       3. log1p-undefine 73.6%

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

       4. rem-exp-log 73.6%

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

       5. associate-+r- 73.6%

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

       6. metadata-eval 73.6%

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

       7. metadata-eval 73.6%

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

   12. Simplified 73.6%

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

   if 1.79999999999999992e-6 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. sqr-neg 99.5%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 96.6%

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

      6. add-sqr-sqrt 96.6%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.6%

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

      10. sqr-neg 1.6%

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

      11. sqrt-unprod 1.6%

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

      12. add-sqr-sqrt 1.6%

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

      13. pow1 1.6%

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

      14. exp-neg 1.6%

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

      15. inv-pow 1.6%

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

      16. pow-prod-up 96.7%

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

      17. metadata-eval 96.7%

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

      18. metadata-eval 96.7%

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

      19. metadata-eval 96.7%

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

   9. Applied egg-rr 49.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\ell \cdot \left(-1 + \left(2 - w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.2% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.38:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.38) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.38) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.38d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.38) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.38:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.38)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.38)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.38], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.38

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 37.4%

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

      3. sqr-neg 37.4%

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

      4. sqrt-unprod 37.4%

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

      5. add-sqr-sqrt 37.4%

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

      6. add-sqr-sqrt 37.4%

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

      7. sqrt-unprod 37.4%

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

      8. add-sqr-sqrt 37.4%

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

      9. sqrt-unprod 37.4%

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

      10. sqr-neg 37.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.6%

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

      17. metadata-eval 98.6%

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

      18. metadata-eval 98.6%

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

      19. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in w around 0 65.9%

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

   6. Taylor expanded in w around 0 19.8%

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

      1. *-rgt-identity 19.8%

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

      2. neg-mul-1 19.8%

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

      3. distribute-rgt-neg-in 19.8%

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

      4. distribute-lft-out 19.8%

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

      5. sub-neg 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in w around inf 19.8%

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

       1. mul-1-neg 19.8%

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

       2. distribute-lft-neg-out 19.8%

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

       3. *-commutative 19.8%

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

   11. Simplified 19.8%

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

   if -0.38 < w

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0 78.2%

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

Alternative 12: 63.9% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(-1 + \left(2 - w\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (+ -1.0 (- 2.0 w))))

C

double code(double w, double l) {
	return l * (-1.0 + (2.0 - w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * ((-1.0d0) + (2.0d0 - w))
end function

Java

public static double code(double w, double l) {
	return l * (-1.0 + (2.0 - w));
}

Python

def code(w, l):
	return l * (-1.0 + (2.0 - w))

Julia

function code(w, l)
	return Float64(l * Float64(-1.0 + Float64(2.0 - w)))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (-1.0 + (2.0 - w));
end

Wolfram

code[w_, l_] := N[(l * N[(-1.0 + N[(2.0 - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 44.7%

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

   2. sqrt-unprod 82.7%

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

   3. sqr-neg 82.7%

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

   4. sqrt-unprod 38.0%

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

   5. add-sqr-sqrt 81.8%

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

   6. add-sqr-sqrt 81.8%

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

   7. sqrt-unprod 81.8%

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

   8. add-sqr-sqrt 38.0%

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

   9. sqrt-unprod 66.6%

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

   10. sqr-neg 66.6%

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

   11. sqrt-unprod 28.6%

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

   12. add-sqr-sqrt 56.9%

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

   13. pow1 56.9%

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

   14. exp-neg 56.9%

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

   15. inv-pow 56.9%

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

   16. pow-prod-up 97.9%

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

   17. metadata-eval 97.9%

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

   18. metadata-eval 97.9%

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

   19. metadata-eval 97.9%

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

4. Applied egg-rr 97.9%

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

5. Taylor expanded in w around 0 74.5%

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

6. Taylor expanded in w around 0 62.6%

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

   1. *-rgt-identity 62.6%

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

   2. neg-mul-1 62.6%

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

   3. distribute-rgt-neg-in 62.6%

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

   4. distribute-lft-out 62.6%

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

   5. sub-neg 62.6%

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

8. Simplified 62.6%

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

   1. expm1-log1p-u 62.1%

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

10. Applied egg-rr 62.1%

    \[\leadsto \ell \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - w\right)\right)} \]
11. Step-by-step derivation

    1. expm1-undefine 62.1%

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

    2. sub-neg 62.1%

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

    3. log1p-undefine 62.1%

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

    4. rem-exp-log 62.6%

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

    5. associate-+r- 62.6%

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

    6. metadata-eval 62.6%

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

    7. metadata-eval 62.6%

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

12. Simplified 62.6%

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

13. Final simplification 62.6%

    \[\leadsto \ell \cdot \left(-1 + \left(2 - w\right)\right) \]
14. Add Preprocessing

Alternative 13: 63.9% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 44.7%

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

   2. sqrt-unprod 82.7%

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

   3. sqr-neg 82.7%

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

   4. sqrt-unprod 38.0%

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

   5. add-sqr-sqrt 81.8%

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

   6. add-sqr-sqrt 81.8%

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

   7. sqrt-unprod 81.8%

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

   8. add-sqr-sqrt 38.0%

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

   9. sqrt-unprod 66.6%

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

   10. sqr-neg 66.6%

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

   11. sqrt-unprod 28.6%

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

   12. add-sqr-sqrt 56.9%

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

   13. pow1 56.9%

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

   14. exp-neg 56.9%

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

   15. inv-pow 56.9%

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

   16. pow-prod-up 97.9%

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

   17. metadata-eval 97.9%

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

   18. metadata-eval 97.9%

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

   19. metadata-eval 97.9%

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

4. Applied egg-rr 97.9%

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

5. Taylor expanded in w around 0 74.5%

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

6. Taylor expanded in w around 0 62.6%

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

   1. *-rgt-identity 62.6%

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

   2. neg-mul-1 62.6%

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

   3. distribute-rgt-neg-in 62.6%

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

   4. distribute-lft-out 62.6%

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

   5. sub-neg 62.6%

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

8. Simplified 62.6%

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

Alternative 14: 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 99.7%

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

3. Taylor expanded in w around 0 58.7%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))