exp-w (used to crash)

Percentage Accurate: 99.5% → 99.3%

Time: 14.1s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% 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: 99.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.35:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.35)
   (/ l (exp w))
   (/
    (pow l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))
    (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.35) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-2.35d0)) then
        tmp = l / exp(w)
    else
        tmp = (l ** (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))) / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -2.35) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.35:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.35)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))))) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -2.35)
		tmp = l / exp(w);
	else
		tmp = (l ^ (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.35], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -2.35000000000000009

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 61.5%

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

      3. sqr-neg 61.5%

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

      4. sqrt-unprod 61.5%

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

      5. add-sqr-sqrt 61.5%

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

      6. add-sqr-sqrt 61.5%

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

      7. sqrt-unprod 61.5%

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

      8. add-sqr-sqrt 61.5%

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

      9. sqrt-unprod 61.5%

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

      10. sqr-neg 61.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in l around 0 100.0%

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

   if -2.35000000000000009 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in w around 0 99.2%

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

      1. *-commutative 99.2%

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

   10. Simplified 99.2%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. exp-neg 99.4%

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

   2. remove-double-neg 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. remove-double-neg 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in l around inf 94.6%

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

   1. mul-1-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. distribute-lft-neg-in 94.6%

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

   4. log-rec 94.6%

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

   5. remove-double-div 94.6%

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

7. Simplified 94.6%

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

   1. pow-to-exp 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. add-sqr-sqrt 99.4%

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

   4. times-frac 99.4%

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

9. Applied egg-rr 99.4%

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

    1. associate-*l/ 99.4%

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

    2. *-lft-identity 99.4%

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

11. Simplified 99.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

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(Float64(-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 99.4%

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

3. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. exp-neg 99.4%

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

   2. remove-double-neg 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. remove-double-neg 99.4%

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

3. Simplified 99.4%

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

Alternative 5: 98.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(1 + w \cdot 0.5\right)\\ \mathbf{if}\;w \leq -3.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ 1.0 (* w 0.5))))))
   (if (<= w -3.9e+20) (/ l (exp w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * 0.5)));
	double tmp;
	if (w <= -3.9e+20) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (w * (1.0d0 + (w * 0.5d0)))
    if (w <= (-3.9d+20)) then
        tmp = l / exp(w)
    else
        tmp = (l ** t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * 0.5)));
	double tmp;
	if (w <= -3.9e+20) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, t_0) / t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 1.0 + (w * (1.0 + (w * 0.5)))
	tmp = 0
	if w <= -3.9e+20:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, t_0) / t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5))))
	tmp = 0.0
	if (w <= -3.9e+20)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 1.0 + (w * (1.0 + (w * 0.5)));
	tmp = 0.0;
	if (w <= -3.9e+20)
		tmp = l / exp(w);
	else
		tmp = (l ^ t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -3.9e+20], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -3.9e20

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 60.9%

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

      3. sqr-neg 60.9%

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

      4. sqrt-unprod 60.9%

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

      5. add-sqr-sqrt 60.9%

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

      6. add-sqr-sqrt 60.9%

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

      7. sqrt-unprod 60.9%

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

      8. add-sqr-sqrt 60.9%

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

      9. sqrt-unprod 60.9%

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

      10. sqr-neg 60.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in l around 0 100.0%

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

   if -3.9e20 < w

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in w around 0 99.0%

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

      1. *-commutative 98.7%

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

   10. Simplified 99.0%

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

Alternative 6: 99.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 61.5%

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

      3. sqr-neg 61.5%

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

      4. sqrt-unprod 61.5%

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

      5. add-sqr-sqrt 61.5%

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

      6. add-sqr-sqrt 61.5%

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

      7. sqrt-unprod 61.5%

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

      8. add-sqr-sqrt 61.5%

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

      9. sqrt-unprod 61.5%

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

      10. sqr-neg 61.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in l around 0 100.0%

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

   if -1 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0 98.8%

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

      1. *-commutative 99.2%

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

   10. Simplified 98.8%

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

Alternative 7: 97.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. exp-neg 99.4%

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

   2. remove-double-neg 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. remove-double-neg 99.4%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 46.0%

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

   2. sqrt-unprod 88.4%

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

   3. sqr-neg 88.4%

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

   4. sqrt-unprod 42.4%

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

   5. add-sqr-sqrt 87.6%

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

   6. add-sqr-sqrt 87.6%

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

   7. sqrt-unprod 87.6%

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

   8. add-sqr-sqrt 42.4%

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

   9. sqrt-unprod 69.6%

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

   10. sqr-neg 69.6%

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

   11. sqrt-unprod 27.2%

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

   12. add-sqr-sqrt 54.0%

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

   13. pow1 54.0%

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

   14. exp-neg 54.0%

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

   15. inv-pow 54.0%

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

   16. pow-prod-up 97.4%

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

   17. metadata-eval 97.4%

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

   18. metadata-eval 97.4%

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

   19. metadata-eval 97.4%

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

6. Applied egg-rr 97.4%

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

7. Taylor expanded in l around 0 97.4%

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

Alternative 8: 90.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.135:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.135)
   (+ l (* l (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666)))))))
   (* l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.135) {
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = l * 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 <= 0.135d0) then
        tmp = l + (l * (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0)))))))
    else
        tmp = l * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.135) {
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = l * 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.135:
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))))
	else:
		tmp = l * 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.135)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))));
	else
		tmp = Float64(l * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.135)
		tmp = l + (l * (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	else
		tmp = l * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.135], N[(l + N[(l * N[(w * N[(-1.0 + N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.13500000000000001

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 86.2%

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

      3. sqr-neg 86.2%

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

      4. sqrt-unprod 52.1%

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

      5. add-sqr-sqrt 85.6%

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

      6. add-sqr-sqrt 85.6%

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

      7. sqrt-unprod 85.6%

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

      8. add-sqr-sqrt 52.1%

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

      9. sqrt-unprod 85.7%

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

      10. sqr-neg 85.7%

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

      11. sqrt-unprod 33.5%

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

      12. add-sqr-sqrt 66.5%

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

      13. pow1 66.5%

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

      14. exp-neg 66.5%

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

      15. inv-pow 66.5%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in w around 0 89.2%

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

   8. Taylor expanded in l around 0 92.2%

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

   if 0.13500000000000001 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.9%

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

      6. add-sqr-sqrt 95.9%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. distribute-rgt-neg-in 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.1%

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

       3. sqr-neg 3.1%

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

       4. sqrt-unprod 3.6%

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

       5. add-sqr-sqrt 3.6%

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

       6. add-log-exp 3.7%

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

       7. pow1 3.7%

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

       8. pow1 3.7%

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

       9. add-sqr-sqrt 3.7%

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

       10. sqrt-unprod 3.7%

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

       11. sqr-neg 3.7%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.6%

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

       14. add-sqr-sqrt 1.6%

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

       15. sqrt-unprod 1.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 0.1%

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

       18. sqr-neg 0.1%

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

       19. sqrt-unprod 0.1%

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

       20. add-sqr-sqrt 0.1%

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

       21. pow1 0.1%

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

       22. exp-neg 0.1%

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

       23. inv-pow 0.1%

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

       24. pow-prod-up 97.9%

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

       25. metadata-eval 97.9%

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

       26. metadata-eval 97.9%

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

   14. Applied egg-rr 97.9%

       \[\leadsto \ell \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.135:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.4% accurate, 16.9× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	} else {
		tmp = l * 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 * (w * (l * (-0.16666666666666666d0)))) - l))
    else
        tmp = l * 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 * (w * (l * -0.16666666666666666))) - l));
	} else {
		tmp = l * 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.13:
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l))
	else:
		tmp = l * 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(w * Float64(l * -0.16666666666666666))) - l)));
	else
		tmp = Float64(l * 0.0);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < 0.13

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 86.2%

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

      3. sqr-neg 86.2%

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

      4. sqrt-unprod 52.1%

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

      5. add-sqr-sqrt 85.6%

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

      6. add-sqr-sqrt 85.6%

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

      7. sqrt-unprod 85.6%

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

      8. add-sqr-sqrt 52.1%

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

      9. sqrt-unprod 85.7%

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

      10. sqr-neg 85.7%

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

      11. sqrt-unprod 33.5%

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

      12. add-sqr-sqrt 66.5%

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

      13. pow1 66.5%

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

      14. exp-neg 66.5%

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

      15. inv-pow 66.5%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in w around 0 89.2%

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

   8. Taylor expanded in w around inf 89.2%

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

      1. distribute-rgt-out 89.2%

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

      2. metadata-eval 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \ell \cdot \color{blue}{-0.3333333333333333}\right)\right)\right) - \ell\right) \]

      3. +-commutative 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot -0.3333333333333333 + -1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)}\right)\right) - \ell\right) \]

      4. mul-1-neg 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot -0.3333333333333333 + \color{blue}{\left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)}\right)\right)\right) - \ell\right) \]

      5. distribute-rgt-out 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot -0.3333333333333333 + \left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right)\right)\right)\right) - \ell\right) \]

      6. metadata-eval 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot -0.3333333333333333 + \left(-\ell \cdot \color{blue}{-0.5}\right)\right)\right)\right) - \ell\right) \]

      7. fma-undefine 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\mathsf{fma}\left(\ell, -0.3333333333333333, -\ell \cdot -0.5\right)}\right)\right) - \ell\right) \]

      8. neg-mul-1 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \color{blue}{\left(-w \cdot \mathsf{fma}\left(\ell, -0.3333333333333333, -\ell \cdot -0.5\right)\right)} - \ell\right) \]

      9. distribute-rgt-neg-in 89.2%

         \[\leadsto \ell + w \cdot \left(w \cdot \color{blue}{\left(w \cdot \left(-\mathsf{fma}\left(\ell, -0.3333333333333333, -\ell \cdot -0.5\right)\right)\right)} - \ell\right) \]

      10. fma-undefine 89.2%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-\color{blue}{\left(\ell \cdot -0.3333333333333333 + \left(-\ell \cdot -0.5\right)\right)}\right)\right) - \ell\right) \]

      11. distribute-rgt-neg-in 89.2%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-\left(\ell \cdot -0.3333333333333333 + \color{blue}{\ell \cdot \left(--0.5\right)}\right)\right)\right) - \ell\right) \]

      12. metadata-eval 89.2%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-\left(\ell \cdot -0.3333333333333333 + \ell \cdot \color{blue}{0.5}\right)\right)\right) - \ell\right) \]

      13. distribute-lft-out 89.2%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-\color{blue}{\ell \cdot \left(-0.3333333333333333 + 0.5\right)}\right)\right) - \ell\right) \]

      14. metadata-eval 89.2%

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

      15. distribute-rgt-neg-in 89.2%

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

      16. metadata-eval 89.2%

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

   10. Simplified 89.2%

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

   if 0.13 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.9%

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

      6. add-sqr-sqrt 95.9%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. distribute-rgt-neg-in 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.1%

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

       3. sqr-neg 3.1%

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

       4. sqrt-unprod 3.6%

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

       5. add-sqr-sqrt 3.6%

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

       6. add-log-exp 3.7%

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

       7. pow1 3.7%

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

       8. pow1 3.7%

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

       9. add-sqr-sqrt 3.7%

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

       10. sqrt-unprod 3.7%

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

       11. sqr-neg 3.7%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.6%

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

       14. add-sqr-sqrt 1.6%

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

       15. sqrt-unprod 1.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 0.1%

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

       18. sqr-neg 0.1%

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

       19. sqrt-unprod 0.1%

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

       20. add-sqr-sqrt 0.1%

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

       21. pow1 0.1%

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

       22. exp-neg 0.1%

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

       23. inv-pow 0.1%

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

       24. pow-prod-up 97.9%

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

       25. metadata-eval 97.9%

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

       26. metadata-eval 97.9%

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

   14. Applied egg-rr 97.9%

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

Alternative 10: 84.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.115) (+ l (* w (- (* w (* l 0.5)) l))) (* l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.115) {
		tmp = l + (w * ((w * (l * 0.5)) - l));
	} else {
		tmp = l * 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 <= 0.115d0) then
        tmp = l + (w * ((w * (l * 0.5d0)) - l))
    else
        tmp = l * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.115) {
		tmp = l + (w * ((w * (l * 0.5)) - l));
	} else {
		tmp = l * 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.115:
		tmp = l + (w * ((w * (l * 0.5)) - l))
	else:
		tmp = l * 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.115)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(l * 0.5)) - l)));
	else
		tmp = Float64(l * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.115)
		tmp = l + (w * ((w * (l * 0.5)) - l));
	else
		tmp = l * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.115], N[(l + N[(w * N[(N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.115000000000000005

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 86.2%

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

      3. sqr-neg 86.2%

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

      4. sqrt-unprod 52.1%

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

      5. add-sqr-sqrt 85.6%

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

      6. add-sqr-sqrt 85.6%

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

      7. sqrt-unprod 85.6%

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

      8. add-sqr-sqrt 52.1%

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

      9. sqrt-unprod 85.7%

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

      10. sqr-neg 85.7%

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

      11. sqrt-unprod 33.5%

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

      12. add-sqr-sqrt 66.5%

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

      13. pow1 66.5%

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

      14. exp-neg 66.5%

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

      15. inv-pow 66.5%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in l around 0 97.7%

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

   8. Taylor expanded in w around 0 86.7%

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

      1. mul-1-neg 86.7%

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

      2. distribute-rgt-out 86.7%

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

      3. metadata-eval 86.7%

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

      4. distribute-rgt-neg-out 86.7%

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

      5. distribute-rgt-neg-in 86.7%

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

      6. metadata-eval 86.7%

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

   10. Simplified 86.7%

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

   if 0.115000000000000005 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.9%

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

      6. add-sqr-sqrt 95.9%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. distribute-rgt-neg-in 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.1%

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

       3. sqr-neg 3.1%

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

       4. sqrt-unprod 3.6%

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

       5. add-sqr-sqrt 3.6%

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

       6. add-log-exp 3.7%

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

       7. pow1 3.7%

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

       8. pow1 3.7%

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

       9. add-sqr-sqrt 3.7%

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

       10. sqrt-unprod 3.7%

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

       11. sqr-neg 3.7%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.6%

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

       14. add-sqr-sqrt 1.6%

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

       15. sqrt-unprod 1.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 0.1%

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

       18. sqr-neg 0.1%

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

       19. sqrt-unprod 0.1%

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

       20. add-sqr-sqrt 0.1%

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

       21. pow1 0.1%

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

       22. exp-neg 0.1%

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

       23. inv-pow 0.1%

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

       24. pow-prod-up 97.9%

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

       25. metadata-eval 97.9%

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

       26. metadata-eval 97.9%

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

   14. Applied egg-rr 97.9%

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

Alternative 11: 77.5% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -495:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{elif}\;w \leq 0.28:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -495.0) (* w (- l)) (if (<= w 0.28) l (* l 0.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -495.0) {
		tmp = w * -l;
	} else if (w <= 0.28) {
		tmp = l;
	} else {
		tmp = l * 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 <= (-495.0d0)) then
        tmp = w * -l
    else if (w <= 0.28d0) then
        tmp = l
    else
        tmp = l * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -495.0) {
		tmp = w * -l;
	} else if (w <= 0.28) {
		tmp = l;
	} else {
		tmp = l * 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -495.0:
		tmp = w * -l
	elif w <= 0.28:
		tmp = l
	else:
		tmp = l * 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -495.0)
		tmp = Float64(w * Float64(-l));
	elseif (w <= 0.28)
		tmp = l;
	else
		tmp = Float64(l * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -495.0)
		tmp = w * -l;
	elseif (w <= 0.28)
		tmp = l;
	else
		tmp = l * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -495.0], N[(w * (-l)), $MachinePrecision], If[LessEqual[w, 0.28], l, N[(l * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -495

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 61.5%

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

      3. sqr-neg 61.5%

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

      4. sqrt-unprod 61.5%

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

      5. add-sqr-sqrt 61.5%

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

      6. add-sqr-sqrt 61.5%

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

      7. sqrt-unprod 61.5%

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

      8. add-sqr-sqrt 61.5%

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

      9. sqrt-unprod 61.5%

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

      10. sqr-neg 61.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 34.3%

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

      1. mul-1-neg 34.3%

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

      2. unsub-neg 34.3%

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

   9. Simplified 34.3%

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

   10. Taylor expanded in w around inf 34.3%

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

       1. neg-mul-1 34.3%

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

       2. distribute-rgt-neg-in 34.3%

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

   12. Simplified 34.3%

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

   if -495 < w < 0.28000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0 96.7%

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

   if 0.28000000000000003 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.9%

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

      6. add-sqr-sqrt 95.9%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. distribute-rgt-neg-in 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.1%

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

       3. sqr-neg 3.1%

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

       4. sqrt-unprod 3.6%

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

       5. add-sqr-sqrt 3.6%

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

       6. add-log-exp 3.7%

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

       7. pow1 3.7%

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

       8. pow1 3.7%

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

       9. add-sqr-sqrt 3.7%

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

       10. sqrt-unprod 3.7%

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

       11. sqr-neg 3.7%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.6%

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

       14. add-sqr-sqrt 1.6%

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

       15. sqrt-unprod 1.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 0.1%

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

       18. sqr-neg 0.1%

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

       19. sqrt-unprod 0.1%

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

       20. add-sqr-sqrt 0.1%

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

       21. pow1 0.1%

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

       22. exp-neg 0.1%

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

       23. inv-pow 0.1%

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

       24. pow-prod-up 97.9%

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

       25. metadata-eval 97.9%

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

       26. metadata-eval 97.9%

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

   14. Applied egg-rr 97.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -495:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{elif}\;w \leq 0.28:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.19:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.19) (- l (* l w)) (* l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.19) {
		tmp = l - (l * w);
	} else {
		tmp = l * 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 <= 0.19d0) then
        tmp = l - (l * w)
    else
        tmp = l * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.19) {
		tmp = l - (l * w);
	} else {
		tmp = l * 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.19:
		tmp = l - (l * w)
	else:
		tmp = l * 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.19)
		tmp = Float64(l - Float64(l * w));
	else
		tmp = Float64(l * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.19)
		tmp = l - (l * w);
	else
		tmp = l * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.19], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision], N[(l * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.19

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 86.2%

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

      3. sqr-neg 86.2%

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

      4. sqrt-unprod 52.1%

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

      5. add-sqr-sqrt 85.6%

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

      6. add-sqr-sqrt 85.6%

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

      7. sqrt-unprod 85.6%

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

      8. add-sqr-sqrt 52.1%

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

      9. sqrt-unprod 85.7%

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

      10. sqr-neg 85.7%

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

      11. sqrt-unprod 33.5%

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

      12. add-sqr-sqrt 66.5%

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

      13. pow1 66.5%

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

      14. exp-neg 66.5%

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

      15. inv-pow 66.5%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in w around 0 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

   9. Simplified 77.2%

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

   if 0.19 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.9%

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

      6. add-sqr-sqrt 95.9%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. distribute-rgt-neg-in 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.1%

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

       3. sqr-neg 3.1%

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

       4. sqrt-unprod 3.6%

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

       5. add-sqr-sqrt 3.6%

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

       6. add-log-exp 3.7%

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

       7. pow1 3.7%

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

       8. pow1 3.7%

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

       9. add-sqr-sqrt 3.7%

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

       10. sqrt-unprod 3.7%

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

       11. sqr-neg 3.7%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.6%

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

       14. add-sqr-sqrt 1.6%

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

       15. sqrt-unprod 1.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 0.1%

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

       18. sqr-neg 0.1%

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

       19. sqrt-unprod 0.1%

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

       20. add-sqr-sqrt 0.1%

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

       21. pow1 0.1%

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

       22. exp-neg 0.1%

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

       23. inv-pow 0.1%

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

       24. pow-prod-up 97.9%

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

       25. metadata-eval 97.9%

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

       26. metadata-eval 97.9%

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

   14. Applied egg-rr 97.9%

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

Alternative 13: 70.9% accurate, 38.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.09:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.09) l (* l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.09) {
		tmp = l;
	} else {
		tmp = l * 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 <= 0.09d0) then
        tmp = l
    else
        tmp = l * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.09) {
		tmp = l;
	} else {
		tmp = l * 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.09:
		tmp = l
	else:
		tmp = l * 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.09)
		tmp = l;
	else
		tmp = Float64(l * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.09)
		tmp = l;
	else
		tmp = l * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.09], l, N[(l * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.089999999999999997

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0 67.8%

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

   if 0.089999999999999997 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 95.9%

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

      6. add-sqr-sqrt 95.9%

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

      7. sqrt-unprod 95.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.1%

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

      10. sqr-neg 0.1%

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

      11. sqrt-unprod 0.1%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 96.0%

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

      17. metadata-eval 96.0%

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

      18. metadata-eval 96.0%

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

      19. metadata-eval 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in w around inf 3.5%

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

       1. neg-mul-1 3.5%

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

       2. distribute-rgt-neg-in 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.1%

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

       3. sqr-neg 3.1%

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

       4. sqrt-unprod 3.6%

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

       5. add-sqr-sqrt 3.6%

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

       6. add-log-exp 3.7%

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

       7. pow1 3.7%

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

       8. pow1 3.7%

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

       9. add-sqr-sqrt 3.7%

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

       10. sqrt-unprod 3.7%

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

       11. sqr-neg 3.7%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 1.6%

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

       14. add-sqr-sqrt 1.6%

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

       15. sqrt-unprod 1.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 0.1%

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

       18. sqr-neg 0.1%

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

       19. sqrt-unprod 0.1%

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

       20. add-sqr-sqrt 0.1%

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

       21. pow1 0.1%

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

       22. exp-neg 0.1%

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

       23. inv-pow 0.1%

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

       24. pow-prod-up 97.9%

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

       25. metadata-eval 97.9%

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

       26. metadata-eval 97.9%

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

   14. Applied egg-rr 97.9%

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

Alternative 14: 57.1% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in w around 0 56.1%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))