exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 20.7s

Alternatives: 11

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = -1.464133099440181e+278
  2. l = -1.5086660943530196e+67

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: 99.5% 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, 1.0× 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 99.9%

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

3. Taylor expanded in w around inf

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

   1. *-commutative N/A

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

   2. exp-to-pow N/A

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

   3. remove-double-neg N/A

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

   4. distribute-lft-neg-out N/A

      \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

   5. log-rec N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. +-rgt-identity N/A

      \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

   9. exp-sum N/A

      \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

   10. +-rgt-identity N/A

       \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

   11. unsub-neg N/A

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

   12. div-exp N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

5. Simplified 99.9%

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

Alternative 2: 97.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{elif}\;w \leq 2.4:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68) (/ 1.0 (exp w)) (if (<= w 2.4) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = 1.0 / exp(w);
	} else if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.68d0)) then
        tmp = 1.0d0 / exp(w)
    else if (w <= 2.4d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = 1.0 / Math.exp(w);
	} else if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = 1.0 / math.exp(w)
	elif w <= 2.4:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = Float64(1.0 / exp(w));
	elseif (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = 1.0 / exp(w);
	elseif (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.4], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.680000000000000049

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 99.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 99.1%

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

      1. *-rgt-identity N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \]

      2. exp-neg N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{w}\right)}\right) \]

      4. exp-lowering-exp.f64 99.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(w\right)\right) \]

   6. Applied egg-rr 99.1%

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

   if -0.680000000000000049 < w < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell} \]
   4. Step-by-step derivation

   5. Simplified 97.3%

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

   if 2.39999999999999991 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in w around inf

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

   1. *-commutative N/A

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

   2. exp-to-pow N/A

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

   3. remove-double-neg N/A

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

   4. distribute-lft-neg-out N/A

      \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

   5. log-rec N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. +-rgt-identity N/A

      \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

   9. exp-sum N/A

      \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

   10. +-rgt-identity N/A

       \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

   11. unsub-neg N/A

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

   12. div-exp N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

5. Simplified 99.9%

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

6. Taylor expanded in w around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
7. Step-by-step derivation

8. Simplified 98.3%

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

Alternative 4: 94.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(w \cdot w\right)\\ \mathbf{if}\;w \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq -0.7:\\ \;\;\;\;1 + \frac{\frac{w \cdot \left(1 + -0.015625 \cdot \left(t\_0 \cdot t\_0\right)\right)}{-1 + t\_0 \cdot -0.125}}{1 + \left(w \cdot 0.5\right) \cdot \left(w \cdot 0.5 - -1\right)}\\ \mathbf{elif}\;w \leq 2.4:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (* w w))))
   (if (<= w -5.6e+102)
     (+ 1.0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666))))))
     (if (<= w -0.7)
       (+
        1.0
        (/
         (/ (* w (+ 1.0 (* -0.015625 (* t_0 t_0)))) (+ -1.0 (* t_0 -0.125)))
         (+ 1.0 (* (* w 0.5) (- (* w 0.5) -1.0)))))
       (if (<= w 2.4) l 0.0)))))

C

double code(double w, double l) {
	double t_0 = w * (w * w);
	double tmp;
	if (w <= -5.6e+102) {
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	} else if (w <= -0.7) {
		tmp = 1.0 + (((w * (1.0 + (-0.015625 * (t_0 * t_0)))) / (-1.0 + (t_0 * -0.125))) / (1.0 + ((w * 0.5) * ((w * 0.5) - -1.0))));
	} else if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	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 = w * (w * w)
    if (w <= (-5.6d+102)) then
        tmp = 1.0d0 + (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0))))))
    else if (w <= (-0.7d0)) then
        tmp = 1.0d0 + (((w * (1.0d0 + ((-0.015625d0) * (t_0 * t_0)))) / ((-1.0d0) + (t_0 * (-0.125d0)))) / (1.0d0 + ((w * 0.5d0) * ((w * 0.5d0) - (-1.0d0)))))
    else if (w <= 2.4d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = w * (w * w);
	double tmp;
	if (w <= -5.6e+102) {
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	} else if (w <= -0.7) {
		tmp = 1.0 + (((w * (1.0 + (-0.015625 * (t_0 * t_0)))) / (-1.0 + (t_0 * -0.125))) / (1.0 + ((w * 0.5) * ((w * 0.5) - -1.0))));
	} else if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = w * (w * w)
	tmp = 0
	if w <= -5.6e+102:
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))))
	elif w <= -0.7:
		tmp = 1.0 + (((w * (1.0 + (-0.015625 * (t_0 * t_0)))) / (-1.0 + (t_0 * -0.125))) / (1.0 + ((w * 0.5) * ((w * 0.5) - -1.0))))
	elif w <= 2.4:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(w * Float64(w * w))
	tmp = 0.0
	if (w <= -5.6e+102)
		tmp = Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))))));
	elseif (w <= -0.7)
		tmp = Float64(1.0 + Float64(Float64(Float64(w * Float64(1.0 + Float64(-0.015625 * Float64(t_0 * t_0)))) / Float64(-1.0 + Float64(t_0 * -0.125))) / Float64(1.0 + Float64(Float64(w * 0.5) * Float64(Float64(w * 0.5) - -1.0)))));
	elseif (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = w * (w * w);
	tmp = 0.0;
	if (w <= -5.6e+102)
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	elseif (w <= -0.7)
		tmp = 1.0 + (((w * (1.0 + (-0.015625 * (t_0 * t_0)))) / (-1.0 + (t_0 * -0.125))) / (1.0 + ((w * 0.5) * ((w * 0.5) - -1.0))));
	elseif (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -5.6e+102], 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, -0.7], N[(1.0 + N[(N[(N[(w * N[(1.0 + N[(-0.015625 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(w * 0.5), $MachinePrecision] * N[(N[(w * 0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.4], l, 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if w < -5.60000000000000037e102

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + \color{blue}{w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \color{blue}{\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot w\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   if -5.60000000000000037e102 < w < -0.69999999999999996

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 97.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + \color{blue}{\frac{1}{2} \cdot w}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 4.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   7. Simplified 4.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(-1 + w \cdot \frac{1}{2}\right) \cdot \color{blue}{w}\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{{-1}^{3} + {\left(w \cdot \frac{1}{2}\right)}^{3}}{-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)} \cdot w\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left({-1}^{3} + {\left(w \cdot \frac{1}{2}\right)}^{3}\right) \cdot w}{\color{blue}{-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left({-1}^{3} + {\left(w \cdot \frac{1}{2}\right)}^{3}\right) \cdot w\right), \color{blue}{\left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({-1}^{3} + {\left(w \cdot \frac{1}{2}\right)}^{3}\right), w\right), \left(\color{blue}{-1 \cdot -1} + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-1 + {\left(w \cdot \frac{1}{2}\right)}^{3}\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left({\left(w \cdot \frac{1}{2}\right)}^{3}\right)\right), w\right), \left(\color{blue}{-1} \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left({\left(\frac{1}{2} \cdot w\right)}^{3}\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left({\frac{1}{2}}^{3} \cdot {w}^{3}\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({\frac{1}{2}}^{3}\right), \left({w}^{3}\right)\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \left({w}^{3}\right)\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \left(w \cdot \left(w \cdot w\right)\right)\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(w, \left(w \cdot w\right)\right)\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), w\right), \left(-1 \cdot -1 + \left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), w\right), \left(1 + \left(\color{blue}{\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right)} - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), w\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\left(w \cdot \frac{1}{2}\right) \cdot \left(w \cdot \frac{1}{2}\right) - -1 \cdot \left(w \cdot \frac{1}{2}\right)\right)}\right)\right)\right) \]

      17. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), w\right), \mathsf{+.f64}\left(1, \left(\left(w \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(w \cdot \frac{1}{2} - -1\right)}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), w\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(w \cdot \frac{1}{2}\right), \color{blue}{\left(w \cdot \frac{1}{2} - -1\right)}\right)\right)\right)\right) \]

   9. Applied egg-rr 25.8%

      \[\leadsto 1 + \color{blue}{\frac{\left(-1 + 0.125 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right) \cdot w}{1 + \left(w \cdot 0.5\right) \cdot \left(w \cdot 0.5 - -1\right)}} \]
   10. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1 \cdot -1 - \left(\frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)}{-1 - \frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)} \cdot w\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{2}\right), -1\right)\right)\right)\right)\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot -1 - \left(\frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\right) \cdot w}{-1 - \frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{2}\right), -1\right)\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(-1 \cdot -1 - \left(\frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\right) \cdot w\right), \left(-1 - \frac{1}{8} \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{2}\right), -1\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 53.6%

       \[\leadsto 1 + \frac{\color{blue}{\frac{\left(1 + -0.015625 \cdot \left(\left(w \cdot \left(w \cdot w\right)\right) \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\right) \cdot w}{-1 + \left(w \cdot \left(w \cdot w\right)\right) \cdot -0.125}}}{1 + \left(w \cdot 0.5\right) \cdot \left(w \cdot 0.5 - -1\right)} \]

   if -0.69999999999999996 < w < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell} \]
   4. Step-by-step derivation

   5. Simplified 97.3%

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

   if 2.39999999999999991 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq -0.7:\\ \;\;\;\;1 + \frac{\frac{w \cdot \left(1 + -0.015625 \cdot \left(\left(w \cdot \left(w \cdot w\right)\right) \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\right)}{-1 + \left(w \cdot \left(w \cdot w\right)\right) \cdot -0.125}}{1 + \left(w \cdot 0.5\right) \cdot \left(w \cdot 0.5 - -1\right)}\\ \mathbf{elif}\;w \leq 2.4:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.13:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5 - \left(\ell \cdot w\right) \cdot 0.16666666666666666\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.13)
   (+ l (* w (- (* w (- (* l 0.5) (* (* l w) 0.16666666666666666))) l)))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l + (w * ((w * ((l * 0.5) - ((l * w) * 0.16666666666666666))) - l));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.13d0) then
        tmp = l + (w * ((w * ((l * 0.5d0) - ((l * w) * 0.16666666666666666d0))) - l))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l + (w * ((w * ((l * 0.5) - ((l * w) * 0.16666666666666666))) - l));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.13:
		tmp = l + (w * ((w * ((l * 0.5) - ((l * w) * 0.16666666666666666))) - l))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.13)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(Float64(l * 0.5) - Float64(Float64(l * w) * 0.16666666666666666))) - l)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.13)
		tmp = l + (w * ((w * ((l * 0.5) - ((l * w) * 0.16666666666666666))) - l));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.13], N[(l + N[(w * N[(N[(w * N[(N[(l * 0.5), $MachinePrecision] - N[(N[(l * w), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.13

   1. Initial program 99.9%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. distribute-lft-neg-out N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. unsub-neg N/A

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

      12. div-exp N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 98.3%

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

   9. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right) + \left(\frac{-1}{2} \cdot \ell + \frac{1}{6} \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)} \]

   11. Simplified 86.6%

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

   if 0.13 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3 \cdot 10^{+97}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq 2.4:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3e+97)
   (+ 1.0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666))))))
   (if (<= w 2.4) (+ l (* w (* l (+ -1.0 (* w 0.5))))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3e+97) {
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	} else if (w <= 2.4) {
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-3d+97)) then
        tmp = 1.0d0 + (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0))))))
    else if (w <= 2.4d0) then
        tmp = l + (w * (l * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -3e+97) {
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	} else if (w <= 2.4) {
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3e+97:
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))))
	elif w <= 2.4:
		tmp = l + (w * (l * (-1.0 + (w * 0.5))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3e+97)
		tmp = Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))))));
	elseif (w <= 2.4)
		tmp = Float64(l + Float64(w * Float64(l * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -3e+97)
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	elseif (w <= 2.4)
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3e+97], 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, 2.4], N[(l + N[(w * N[(l * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -2.9999999999999998e97

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + \color{blue}{w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \color{blue}{\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot w\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 96.5%

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

   if -2.9999999999999998e97 < w < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. distribute-lft-neg-out N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. unsub-neg N/A

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

      12. div-exp N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 97.2%

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

   9. Taylor expanded in w around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \color{blue}{\left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\mathsf{neg}\left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right)\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\ell \cdot \left(-1 + \frac{1}{2}\right)\right)\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\ell \cdot \frac{-1}{2}\right)\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \ell\right)\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \ell\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} \cdot \ell\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(w \cdot \frac{1}{2}\right) \cdot \ell + \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot w\right) \cdot \ell + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot w\right) \cdot \ell + -1 \cdot \color{blue}{\ell}\right)\right)\right) \]

       14. distribute-rgt-out N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\ell \cdot \color{blue}{\left(\frac{1}{2} \cdot w + -1\right)}\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\ell \cdot \left(\frac{1}{2} \cdot w + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\ell \cdot \left(\frac{1}{2} \cdot w - \color{blue}{1}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} \cdot w - 1\right)}\right)\right)\right) \]

       18. sub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w + -1\right)\right)\right)\right) \]

       20. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \left(-1 + \color{blue}{\frac{1}{2} \cdot w}\right)\right)\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right)\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(-1, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   11. Simplified 82.8%

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

   if 2.39999999999999991 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.0% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 2.4:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.9e+154)
   (* w (* w 0.5))
   (if (<= w 2.4) (+ l (* w (* l (+ -1.0 (* w 0.5))))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.9e+154) {
		tmp = w * (w * 0.5);
	} else if (w <= 2.4) {
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	} else {
		tmp = 0.0;
	}
	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+154)) then
        tmp = w * (w * 0.5d0)
    else if (w <= 2.4d0) then
        tmp = l + (w * (l * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.9e+154) {
		tmp = w * (w * 0.5);
	} else if (w <= 2.4) {
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.9e+154:
		tmp = w * (w * 0.5)
	elif w <= 2.4:
		tmp = l + (w * (l * (-1.0 + (w * 0.5))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.9e+154)
		tmp = Float64(w * Float64(w * 0.5));
	elseif (w <= 2.4)
		tmp = Float64(l + Float64(w * Float64(l * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.9e+154)
		tmp = w * (w * 0.5);
	elseif (w <= 2.4)
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.9e+154], N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.4], N[(l + N[(w * N[(l * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -1.8999999999999999e154

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + \color{blue}{\frac{1}{2} \cdot w}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right) \]

   10. Simplified 100.0%

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

   if -1.8999999999999999e154 < w < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. distribute-lft-neg-out N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. unsub-neg N/A

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

      12. div-exp N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 97.4%

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

   9. Taylor expanded in w around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \color{blue}{\left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\mathsf{neg}\left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right)\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\ell \cdot \left(-1 + \frac{1}{2}\right)\right)\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\ell \cdot \frac{-1}{2}\right)\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \ell\right)\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \ell\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} \cdot \ell\right) + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(w \cdot \frac{1}{2}\right) \cdot \ell + \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot w\right) \cdot \ell + \left(\mathsf{neg}\left(\ell\right)\right)\right)\right)\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot w\right) \cdot \ell + -1 \cdot \color{blue}{\ell}\right)\right)\right) \]

       14. distribute-rgt-out N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\ell \cdot \color{blue}{\left(\frac{1}{2} \cdot w + -1\right)}\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\ell \cdot \left(\frac{1}{2} \cdot w + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\ell \cdot \left(\frac{1}{2} \cdot w - \color{blue}{1}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} \cdot w - 1\right)}\right)\right)\right) \]

       18. sub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w + -1\right)\right)\right)\right) \]

       20. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \left(-1 + \color{blue}{\frac{1}{2} \cdot w}\right)\right)\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right)\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(-1, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   11. Simplified 81.4%

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

   if 2.39999999999999991 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.8 \cdot 10^{+142}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 0.19:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.8e+142) (* w (* w 0.5)) (if (<= w 0.19) (* l (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.8e+142) {
		tmp = w * (w * 0.5);
	} else if (w <= 0.19) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	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+142)) then
        tmp = w * (w * 0.5d0)
    else if (w <= 0.19d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.8e+142) {
		tmp = w * (w * 0.5);
	} else if (w <= 0.19) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.8e+142:
		tmp = w * (w * 0.5)
	elif w <= 0.19:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.8e+142)
		tmp = Float64(w * Float64(w * 0.5));
	elseif (w <= 0.19)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.8e+142)
		tmp = w * (w * 0.5);
	elseif (w <= 0.19)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.8e+142], N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.19], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -1.8000000000000001e142

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + \color{blue}{\frac{1}{2} \cdot w}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   7. Simplified 97.7%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      6. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right) \]

   10. Simplified 97.7%

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

   if -1.8000000000000001e142 < w < 0.19

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 75.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 75.9%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 79.3%

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

   if 0.19 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.8 \cdot 10^{+142}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 0.19:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.5 \cdot 10^{+65}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 2.4:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3.5e+65) (* w (* w 0.5)) (if (<= w 2.4) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3.5e+65) {
		tmp = w * (w * 0.5);
	} else if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-3.5d+65)) then
        tmp = w * (w * 0.5d0)
    else if (w <= 2.4d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -3.5e+65) {
		tmp = w * (w * 0.5);
	} else if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3.5e+65:
		tmp = w * (w * 0.5)
	elif w <= 2.4:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3.5e+65)
		tmp = Float64(w * Float64(w * 0.5));
	elseif (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -3.5e+65)
		tmp = w * (w * 0.5);
	elseif (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3.5e+65], N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.4], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -3.5000000000000001e65

   1. Initial program 100.0%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\color{blue}{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\color{blue}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}}\right)}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)}\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{\left(0 - \color{blue}{0}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \left({\ell}^{0}\right)\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), 1\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} \cdot w + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + \color{blue}{\frac{1}{2} \cdot w}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 67.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   7. Simplified 67.1%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w, \color{blue}{\left(\frac{1}{2} \cdot w\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w, \left(w \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      6. *-lowering-*.f64 67.1%

         \[\leadsto \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \color{blue}{\frac{1}{2}}\right)\right) \]

   10. Simplified 67.1%

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

   if -3.5000000000000001e65 < w < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell} \]
   4. Step-by-step derivation

   5. Simplified 84.4%

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

   if 2.39999999999999991 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.9% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2.4:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 2.4) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 2.4d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2.4) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2.4:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2.4)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2.4], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell} \]
   4. Step-by-step derivation

   5. Simplified 61.6%

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

   if 2.39999999999999991 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

         \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 16.7% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 0.0)

C

double code(double w, double l) {
	return 0.0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 0.0d0
end function

Java

public static double code(double w, double l) {
	return 0.0;
}

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

function tmp = code(w, l)
	tmp = 0.0;
end

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.9%

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

   1. exp-neg N/A

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

   2. sqr-pow N/A

      \[\leadsto \frac{1}{e^{w}} \cdot \left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2}\right)}}\right) \]

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

   8. mul0-lft N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   11. mul0-lft N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   13. +-inverses N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

   15. flip-- N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

   17. metadata-eval N/A

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

   18. associate-/r/ N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)} \]

   22. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

   23. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{\color{blue}{2}}\right)} \]

4. Applied egg-rr 20.0%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))