exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 23.3s

Alternatives: 14

Speedup: N/A×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. exp-neg 99.7%

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

   2. remove-double-neg 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-lft-identity 99.7%

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

   5. remove-double-neg 99.7%

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

3. Simplified 99.7%

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

Alternative 2: 97.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 41.3%

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

   2. sqrt-unprod 85.0%

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

   3. sqr-neg 85.0%

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

   4. sqrt-unprod 43.7%

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

   5. add-sqr-sqrt 84.7%

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

   6. add-sqr-sqrt 84.7%

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

   7. sqrt-unprod 84.7%

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

   8. add-sqr-sqrt 43.7%

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

   9. sqrt-unprod 68.7%

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

   10. sqr-neg 68.7%

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

   11. sqrt-unprod 25.0%

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

   12. add-sqr-sqrt 52.0%

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

   13. pow1 52.0%

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

   14. exp-neg 52.0%

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

   15. inv-pow 52.0%

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

   16. pow-prod-up 98.4%

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

   17. metadata-eval 98.4%

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

   18. metadata-eval 98.4%

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

   19. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 97.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 41.3%

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

   2. sqrt-unprod 85.0%

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

   3. sqr-neg 85.0%

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

   4. sqrt-unprod 43.7%

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

   5. add-sqr-sqrt 84.7%

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

   6. add-sqr-sqrt 84.7%

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

   7. sqrt-unprod 84.7%

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

   8. add-sqr-sqrt 43.7%

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

   9. sqrt-unprod 68.7%

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

   10. sqr-neg 68.7%

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

   11. sqrt-unprod 25.0%

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

   12. add-sqr-sqrt 52.0%

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

   13. pow1 52.0%

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

   14. exp-neg 52.0%

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

   15. inv-pow 52.0%

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

   16. pow-prod-up 98.4%

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

   17. metadata-eval 98.4%

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

   18. metadata-eval 98.4%

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

   19. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in w around inf 98.4%

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

   1. rem-exp-log 94.0%

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

   2. prod-exp 94.0%

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

   3. sub-neg 94.0%

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

   4. div-exp 94.0%

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

   5. rem-exp-log 98.4%

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

7. Simplified 98.4%

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

Alternative 4: 89.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(\ell \cdot \left(w \cdot -0.16666666666666666 - 0.5\right)\right)\\ t_1 := 0.5 + w \cdot -0.16666666666666666\\ \mathbf{if}\;w \leq -2.95 \cdot 10^{+54}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot t\_1 + -1\right)\right)\\ \mathbf{elif}\;w \leq -0.24:\\ \;\;\;\;\ell + w \cdot \left(\frac{\left(w \cdot \left(\ell \cdot t\_1\right)\right) \cdot t\_0}{t\_0} - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (* l (- (* w -0.16666666666666666) 0.5))))
        (t_1 (+ 0.5 (* w -0.16666666666666666))))
   (if (<= w -2.95e+54)
     (+ l (* l (* w (+ (* w t_1) -1.0))))
     (if (<= w -0.24)
       (+ l (* w (- (/ (* (* w (* l t_1)) t_0) t_0) l)))
       (/ l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))))

C

double code(double w, double l) {
	double t_0 = w * (l * ((w * -0.16666666666666666) - 0.5));
	double t_1 = 0.5 + (w * -0.16666666666666666);
	double tmp;
	if (w <= -2.95e+54) {
		tmp = l + (l * (w * ((w * t_1) + -1.0)));
	} else if (w <= -0.24) {
		tmp = l + (w * ((((w * (l * t_1)) * t_0) / t_0) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = w * (l * ((w * (-0.16666666666666666d0)) - 0.5d0))
    t_1 = 0.5d0 + (w * (-0.16666666666666666d0))
    if (w <= (-2.95d+54)) then
        tmp = l + (l * (w * ((w * t_1) + (-1.0d0))))
    else if (w <= (-0.24d0)) then
        tmp = l + (w * ((((w * (l * t_1)) * t_0) / t_0) - l))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = w * (l * ((w * -0.16666666666666666) - 0.5));
	double t_1 = 0.5 + (w * -0.16666666666666666);
	double tmp;
	if (w <= -2.95e+54) {
		tmp = l + (l * (w * ((w * t_1) + -1.0)));
	} else if (w <= -0.24) {
		tmp = l + (w * ((((w * (l * t_1)) * t_0) / t_0) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = w * (l * ((w * -0.16666666666666666) - 0.5))
	t_1 = 0.5 + (w * -0.16666666666666666)
	tmp = 0
	if w <= -2.95e+54:
		tmp = l + (l * (w * ((w * t_1) + -1.0)))
	elif w <= -0.24:
		tmp = l + (w * ((((w * (l * t_1)) * t_0) / t_0) - l))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	t_0 = Float64(w * Float64(l * Float64(Float64(w * -0.16666666666666666) - 0.5)))
	t_1 = Float64(0.5 + Float64(w * -0.16666666666666666))
	tmp = 0.0
	if (w <= -2.95e+54)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * t_1) + -1.0))));
	elseif (w <= -0.24)
		tmp = Float64(l + Float64(w * Float64(Float64(Float64(Float64(w * Float64(l * t_1)) * t_0) / t_0) - l)));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = w * (l * ((w * -0.16666666666666666) - 0.5));
	t_1 = 0.5 + (w * -0.16666666666666666);
	tmp = 0.0;
	if (w <= -2.95e+54)
		tmp = l + (l * (w * ((w * t_1) + -1.0)));
	elseif (w <= -0.24)
		tmp = l + (w * ((((w * (l * t_1)) * t_0) / t_0) - l));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(l * N[(N[(w * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.95e+54], N[(l + N[(l * N[(w * N[(N[(w * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -0.24], N[(l + N[(w * N[(N[(N[(N[(w * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.9499999999999999e54

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 55.4%

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

      3. sqr-neg 55.4%

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

      4. sqrt-unprod 55.4%

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

      5. add-sqr-sqrt 55.4%

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

      6. add-sqr-sqrt 55.4%

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

      7. sqrt-unprod 55.4%

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

      8. add-sqr-sqrt 55.4%

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

      9. sqrt-unprod 55.4%

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

      10. sqr-neg 55.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0 81.1%

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

   6. Taylor expanded in l around 0 86.9%

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

   if -2.9499999999999999e54 < w < -0.23999999999999999

   1. Initial program 99.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.4%

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

      3. sqr-neg 47.4%

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

      4. sqrt-unprod 47.4%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-sqr-sqrt 47.4%

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

      7. sqrt-unprod 47.4%

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

      8. add-sqr-sqrt 47.4%

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

      9. sqrt-unprod 47.4%

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

      10. sqr-neg 47.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.0%

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

      13. pow1 1.0%

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

      14. exp-neg 1.0%

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

      15. inv-pow 1.0%

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

      16. pow-prod-up 87.7%

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

      17. metadata-eval 87.7%

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

      18. metadata-eval 87.7%

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

      19. metadata-eval 87.7%

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

   4. Applied egg-rr 87.7%

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

   5. Taylor expanded in w around 0 11.7%

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

      1. distribute-lft-in 11.7%

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

      2. flip-+ 15.9%

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

      3. *-commutative 15.9%

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

      4. associate-*l* 15.9%

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

      5. *-commutative 15.9%

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

      6. associate-*l* 15.9%

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

      7. *-commutative 15.9%

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

      8. *-commutative 15.9%

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

      9. *-commutative 15.9%

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

      10. associate-*l* 15.9%

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

      11. *-commutative 15.9%

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

   7. Applied egg-rr 15.9%

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

      1. difference-of-squares 42.5%

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

      2. distribute-lft-out 42.5%

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

      3. +-commutative 42.5%

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

      4. distribute-lft-in 42.5%

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

      5. distribute-lft-out-- 42.5%

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

      6. distribute-lft-out-- 42.5%

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

      7. distribute-lft-out-- 42.5%

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

      8. distribute-lft-out-- 42.5%

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

   9. Simplified 42.5%

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

   if -0.23999999999999999 < w

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 60.0%

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

      2. sqrt-unprod 99.1%

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

      3. sqr-neg 99.1%

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

      4. sqrt-unprod 39.0%

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

      5. add-sqr-sqrt 98.7%

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

      6. add-sqr-sqrt 98.7%

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

      7. sqrt-unprod 98.7%

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

      8. add-sqr-sqrt 39.0%

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

      9. sqrt-unprod 75.5%

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

      10. sqr-neg 75.5%

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

      11. sqrt-unprod 36.4%

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

      12. add-sqr-sqrt 75.5%

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

      13. pow1 75.5%

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

      14. exp-neg 75.5%

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

      15. inv-pow 75.5%

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

      16. pow-prod-up 98.8%

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

      17. metadata-eval 98.8%

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

      18. metadata-eval 98.8%

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

      19. metadata-eval 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in w around inf 98.8%

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

      1. rem-exp-log 92.4%

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

      2. prod-exp 92.4%

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

      3. sub-neg 92.4%

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

      4. div-exp 92.4%

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

      5. rem-exp-log 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0 95.4%

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

      1. *-commutative 95.4%

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

   10. Simplified 95.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.95 \cdot 10^{+54}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;w \leq -0.24:\\ \;\;\;\;\ell + w \cdot \left(\frac{\left(w \cdot \left(\ell \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \left(w \cdot \left(\ell \cdot \left(w \cdot -0.16666666666666666 - 0.5\right)\right)\right)}{w \cdot \left(\ell \cdot \left(w \cdot -0.16666666666666666 - 0.5\right)\right)} - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2e-7)
   (* l (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2e-7) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 2d-7) then
        tmp = l * (1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2e-7) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2e-7:
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2e-7)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2e-7)
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2e-7], N[(l * N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.9999999999999999e-7

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 30.1%

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

      2. sqrt-unprod 82.1%

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

      3. sqr-neg 82.1%

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

      4. sqrt-unprod 52.0%

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

      5. add-sqr-sqrt 81.8%

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

      6. add-sqr-sqrt 81.8%

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

      7. sqrt-unprod 81.8%

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

      8. add-sqr-sqrt 52.0%

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

      9. sqrt-unprod 81.8%

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

      10. sqr-neg 81.8%

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

      11. sqrt-unprod 29.8%

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

      12. add-sqr-sqrt 61.9%

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

      13. pow1 61.9%

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

      14. exp-neg 61.9%

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

      15. inv-pow 61.9%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 88.9%

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

   if 1.9999999999999999e-7 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. rem-exp-log 100.0%

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

      2. prod-exp 100.0%

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

      3. sub-neg 100.0%

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

      4. div-exp 100.0%

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

      5. rem-exp-log 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 85.5%

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

      1. *-commutative 85.5%

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

   10. Simplified 85.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2e-7)
   (+ l (* l (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2e-7) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 2d-7) then
        tmp = l + (l * (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2e-7) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2e-7:
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2e-7)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2e-7)
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2e-7], N[(l + N[(l * N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.9999999999999999e-7

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 30.1%

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

      2. sqrt-unprod 82.1%

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

      3. sqr-neg 82.1%

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

      4. sqrt-unprod 52.0%

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

      5. add-sqr-sqrt 81.8%

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

      6. add-sqr-sqrt 81.8%

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

      7. sqrt-unprod 81.8%

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

      8. add-sqr-sqrt 52.0%

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

      9. sqrt-unprod 81.8%

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

      10. sqr-neg 81.8%

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

      11. sqrt-unprod 29.8%

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

      12. add-sqr-sqrt 61.9%

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

      13. pow1 61.9%

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

      14. exp-neg 61.9%

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

      15. inv-pow 61.9%

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

      16. pow-prod-up 98.1%

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

      17. metadata-eval 98.1%

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

      18. metadata-eval 98.1%

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

      19. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in w around 0 87.1%

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

   6. Taylor expanded in l around 0 88.9%

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

   if 1.9999999999999999e-7 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. rem-exp-log 100.0%

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

      2. prod-exp 100.0%

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

      3. sub-neg 100.0%

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

      4. div-exp 100.0%

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

      5. rem-exp-log 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 85.5%

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

      1. *-commutative 85.5%

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

   10. Simplified 85.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -12.2:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -12.2)
   (+ l (* l (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -12.2) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-12.2d0)) then
        tmp = l + (l * (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -12.2) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -12.2:
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -12.2)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -12.2)
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -12.2], N[(l + N[(l * N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(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 < -12.199999999999999

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 55.1%

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

      3. sqr-neg 55.1%

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

      4. sqrt-unprod 55.1%

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

      5. add-sqr-sqrt 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. sqrt-unprod 55.1%

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

      8. add-sqr-sqrt 55.1%

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

      9. sqrt-unprod 55.1%

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

      10. sqr-neg 55.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0 69.6%

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

   6. Taylor expanded in l around 0 74.4%

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

   if -12.199999999999999 < w

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 59.4%

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

      2. sqrt-unprod 98.0%

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

      3. sqr-neg 98.0%

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

      4. sqrt-unprod 38.7%

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

      5. add-sqr-sqrt 97.7%

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

      6. add-sqr-sqrt 97.7%

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

      7. sqrt-unprod 97.7%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 74.7%

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

      10. sqr-neg 74.7%

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

      11. sqrt-unprod 36.0%

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

      12. add-sqr-sqrt 74.7%

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

      13. pow1 74.7%

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

      14. exp-neg 74.7%

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

      15. inv-pow 74.7%

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

      16. pow-prod-up 97.8%

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

      17. metadata-eval 97.8%

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

      18. metadata-eval 97.8%

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

      19. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in w around inf 97.8%

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

      1. rem-exp-log 91.4%

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

      2. prod-exp 91.4%

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

      3. sub-neg 91.4%

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

      4. div-exp 91.4%

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

      5. rem-exp-log 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in w around 0 93.5%

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

      1. *-commutative 93.5%

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

   10. Simplified 93.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -12.2:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.4:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.4)
   (* l (+ 1.0 (* w (+ -1.0 (* w 0.5)))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.4) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.4d0)) then
        tmp = l * (1.0d0 + (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.4) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.4:
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.4)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.4)
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.4], N[(l * N[(1.0 + N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.3999999999999999

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 55.1%

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

      3. sqr-neg 55.1%

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

      4. sqrt-unprod 55.1%

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

      5. add-sqr-sqrt 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. sqrt-unprod 55.1%

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

      8. add-sqr-sqrt 55.1%

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

      9. sqrt-unprod 55.1%

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

      10. sqr-neg 55.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0 64.5%

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

   if -1.3999999999999999 < w

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 59.4%

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

      2. sqrt-unprod 98.0%

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

      3. sqr-neg 98.0%

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

      4. sqrt-unprod 38.7%

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

      5. add-sqr-sqrt 97.7%

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

      6. add-sqr-sqrt 97.7%

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

      7. sqrt-unprod 97.7%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 74.7%

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

      10. sqr-neg 74.7%

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

      11. sqrt-unprod 36.0%

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

      12. add-sqr-sqrt 74.7%

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

      13. pow1 74.7%

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

      14. exp-neg 74.7%

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

      15. inv-pow 74.7%

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

      16. pow-prod-up 97.8%

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

      17. metadata-eval 97.8%

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

      18. metadata-eval 97.8%

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

      19. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in w around inf 97.8%

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

      1. rem-exp-log 91.4%

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

      2. prod-exp 91.4%

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

      3. sub-neg 91.4%

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

      4. div-exp 91.4%

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

      5. rem-exp-log 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in w around 0 93.5%

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

      1. *-commutative 93.5%

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

   10. Simplified 93.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.4:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -7.3:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -7.3)
   (- l (* w (+ l (* w (* l -0.5)))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -7.3) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-7.3d0)) then
        tmp = l - (w * (l + (w * (l * (-0.5d0)))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -7.3) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -7.3:
		tmp = l - (w * (l + (w * (l * -0.5))))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -7.3)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(l * -0.5)))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -7.3)
		tmp = l - (w * (l + (w * (l * -0.5))));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -7.3], N[(l - N[(w * N[(l + N[(w * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -7.29999999999999982

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 55.1%

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

      3. sqr-neg 55.1%

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

      4. sqrt-unprod 55.1%

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

      5. add-sqr-sqrt 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. sqrt-unprod 55.1%

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

      8. add-sqr-sqrt 55.1%

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

      9. sqrt-unprod 55.1%

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

      10. sqr-neg 55.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. pow1 0.0%

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

      14. exp-neg 0.0%

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

      15. inv-pow 0.0%

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

      16. pow-prod-up 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around inf 100.0%

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

      1. rem-exp-log 100.0%

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

      2. prod-exp 100.0%

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

      3. sub-neg 100.0%

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

      4. div-exp 100.0%

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

      5. rem-exp-log 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 56.1%

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

      1. distribute-rgt-out 56.1%

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

      2. metadata-eval 56.1%

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

      3. metadata-eval 56.1%

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

      4. distribute-rgt-neg-in 56.1%

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

      5. associate-*r* 56.1%

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

      6. neg-mul-1 56.1%

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

      7. distribute-rgt-neg-in 56.1%

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

      8. metadata-eval 56.1%

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

   10. Simplified 56.1%

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

   if -7.29999999999999982 < w

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 59.4%

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

      2. sqrt-unprod 98.0%

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

      3. sqr-neg 98.0%

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

      4. sqrt-unprod 38.7%

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

      5. add-sqr-sqrt 97.7%

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

      6. add-sqr-sqrt 97.7%

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

      7. sqrt-unprod 97.7%

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

      8. add-sqr-sqrt 38.7%

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

      9. sqrt-unprod 74.7%

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

      10. sqr-neg 74.7%

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

      11. sqrt-unprod 36.0%

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

      12. add-sqr-sqrt 74.7%

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

      13. pow1 74.7%

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

      14. exp-neg 74.7%

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

      15. inv-pow 74.7%

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

      16. pow-prod-up 97.8%

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

      17. metadata-eval 97.8%

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

      18. metadata-eval 97.8%

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

      19. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in w around inf 97.8%

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

      1. rem-exp-log 91.4%

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

      2. prod-exp 91.4%

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

      3. sub-neg 91.4%

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

      4. div-exp 91.4%

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

      5. rem-exp-log 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in w around 0 93.5%

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

      1. *-commutative 93.5%

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

   10. Simplified 93.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -7.3:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.122:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.122) (* l (- w)) (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.122) {
		tmp = l * -w;
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.122d0)) then
        tmp = l * -w
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.122) {
		tmp = l * -w;
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.122:
		tmp = l * -w
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.122)
		tmp = Float64(l * Float64(-w));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.122)
		tmp = l * -w;
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.122], N[(l * (-w)), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.122

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 53.9%

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

      5. add-sqr-sqrt 53.9%

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

      6. add-sqr-sqrt 53.9%

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

      7. sqrt-unprod 53.9%

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

      8. add-sqr-sqrt 53.9%

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

      9. sqrt-unprod 53.9%

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

      10. sqr-neg 53.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in w around inf 97.7%

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

      1. rem-exp-log 97.7%

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

      2. prod-exp 97.7%

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

      3. sub-neg 97.7%

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

      4. div-exp 97.7%

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

      5. rem-exp-log 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in w around 0 32.1%

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

      1. *-rgt-identity 32.1%

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

      2. neg-mul-1 32.1%

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

      3. distribute-rgt-neg-in 32.1%

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

      4. distribute-lft-out 32.1%

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

      5. sub-neg 32.1%

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

   10. Simplified 32.1%

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

   11. Taylor expanded in w around inf 32.1%

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

       1. associate-*r* 32.1%

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

       2. mul-1-neg 32.1%

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

   13. Simplified 32.1%

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

   if -0.122 < w

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 60.0%

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

      2. sqrt-unprod 99.1%

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

      3. sqr-neg 99.1%

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

      4. sqrt-unprod 39.0%

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

      5. add-sqr-sqrt 98.7%

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

      6. add-sqr-sqrt 98.7%

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

      7. sqrt-unprod 98.7%

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

      8. add-sqr-sqrt 39.0%

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

      9. sqrt-unprod 75.5%

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

      10. sqr-neg 75.5%

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

      11. sqrt-unprod 36.4%

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

      12. add-sqr-sqrt 75.5%

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

      13. pow1 75.5%

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

      14. exp-neg 75.5%

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

      15. inv-pow 75.5%

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

      16. pow-prod-up 98.8%

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

      17. metadata-eval 98.8%

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

      18. metadata-eval 98.8%

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

      19. metadata-eval 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in w around inf 98.8%

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

      1. rem-exp-log 92.4%

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

      2. prod-exp 92.4%

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

      3. sub-neg 92.4%

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

      4. div-exp 92.4%

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

      5. rem-exp-log 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0 94.5%

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

      1. *-commutative 94.5%

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

   10. Simplified 94.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.122:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.04:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.04) (* l (- w)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.04) {
		tmp = l * -w;
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.04d0)) then
        tmp = l * -w
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.04) {
		tmp = l * -w;
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.04:
		tmp = l * -w
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.04)
		tmp = Float64(l * Float64(-w));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.04)
		tmp = l * -w;
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.04], N[(l * (-w)), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.0400000000000000008

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 53.9%

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

      5. add-sqr-sqrt 53.9%

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

      6. add-sqr-sqrt 53.9%

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

      7. sqrt-unprod 53.9%

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

      8. add-sqr-sqrt 53.9%

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

      9. sqrt-unprod 53.9%

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

      10. sqr-neg 53.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in w around inf 97.7%

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

      1. rem-exp-log 97.7%

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

      2. prod-exp 97.7%

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

      3. sub-neg 97.7%

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

      4. div-exp 97.7%

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

      5. rem-exp-log 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in w around 0 32.1%

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

      1. *-rgt-identity 32.1%

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

      2. neg-mul-1 32.1%

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

      3. distribute-rgt-neg-in 32.1%

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

      4. distribute-lft-out 32.1%

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

      5. sub-neg 32.1%

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

   10. Simplified 32.1%

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

   11. Taylor expanded in w around inf 32.1%

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

       1. associate-*r* 32.1%

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

       2. mul-1-neg 32.1%

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

   13. Simplified 32.1%

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

   if -0.0400000000000000008 < w

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 60.0%

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

      2. sqrt-unprod 99.1%

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

      3. sqr-neg 99.1%

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

      4. sqrt-unprod 39.0%

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

      5. add-sqr-sqrt 98.7%

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

      6. add-sqr-sqrt 98.7%

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

      7. sqrt-unprod 98.7%

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

      8. add-sqr-sqrt 39.0%

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

      9. sqrt-unprod 75.5%

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

      10. sqr-neg 75.5%

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

      11. sqrt-unprod 36.4%

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

      12. add-sqr-sqrt 75.5%

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

      13. pow1 75.5%

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

      14. exp-neg 75.5%

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

      15. inv-pow 75.5%

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

      16. pow-prod-up 98.8%

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

      17. metadata-eval 98.8%

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

      18. metadata-eval 98.8%

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

      19. metadata-eval 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in w around inf 98.8%

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

      1. rem-exp-log 92.4%

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

      2. prod-exp 92.4%

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

      3. sub-neg 92.4%

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

      4. div-exp 92.4%

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

      5. rem-exp-log 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0 89.7%

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

      1. +-commutative 89.7%

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

   10. Simplified 89.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.04:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.011:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.011) (* l (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.011) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.011d0)) then
        tmp = l * -w
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.011) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.011:
		tmp = l * -w
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.011)
		tmp = Float64(l * Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.011)
		tmp = l * -w;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.011], N[(l * (-w)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.010999999999999999

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 53.9%

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

      5. add-sqr-sqrt 53.9%

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

      6. add-sqr-sqrt 53.9%

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

      7. sqrt-unprod 53.9%

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

      8. add-sqr-sqrt 53.9%

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

      9. sqrt-unprod 53.9%

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

      10. sqr-neg 53.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.2%

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

      13. pow1 0.2%

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

      14. exp-neg 0.2%

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

      15. inv-pow 0.2%

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

      16. pow-prod-up 97.7%

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

      17. metadata-eval 97.7%

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

      18. metadata-eval 97.7%

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

      19. metadata-eval 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in w around inf 97.7%

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

      1. rem-exp-log 97.7%

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

      2. prod-exp 97.7%

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

      3. sub-neg 97.7%

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

      4. div-exp 97.7%

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

      5. rem-exp-log 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in w around 0 32.1%

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

      1. *-rgt-identity 32.1%

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

      2. neg-mul-1 32.1%

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

      3. distribute-rgt-neg-in 32.1%

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

      4. distribute-lft-out 32.1%

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

      5. sub-neg 32.1%

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

   10. Simplified 32.1%

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

   11. Taylor expanded in w around inf 32.1%

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

       1. associate-*r* 32.1%

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

       2. mul-1-neg 32.1%

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

   13. Simplified 32.1%

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

   if -0.010999999999999999 < w

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0 76.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.011:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.6% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-sqr-sqrt 41.3%

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

   2. sqrt-unprod 85.0%

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

   3. sqr-neg 85.0%

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

   4. sqrt-unprod 43.7%

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

   5. add-sqr-sqrt 84.7%

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

   6. add-sqr-sqrt 84.7%

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

   7. sqrt-unprod 84.7%

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

   8. add-sqr-sqrt 43.7%

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

   9. sqrt-unprod 68.7%

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

   10. sqr-neg 68.7%

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

   11. sqrt-unprod 25.0%

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

   12. add-sqr-sqrt 52.0%

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

   13. pow1 52.0%

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

   14. exp-neg 52.0%

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

   15. inv-pow 52.0%

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

   16. pow-prod-up 98.4%

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

   17. metadata-eval 98.4%

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

   18. metadata-eval 98.4%

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

   19. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in w around inf 98.4%

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

   1. rem-exp-log 94.0%

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

   2. prod-exp 94.0%

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

   3. sub-neg 94.0%

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

   4. div-exp 94.0%

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

   5. rem-exp-log 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in w around 0 62.5%

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

   1. *-rgt-identity 62.5%

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

   2. neg-mul-1 62.5%

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

   3. distribute-rgt-neg-in 62.5%

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

   4. distribute-lft-out 62.5%

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

   5. sub-neg 62.5%

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

10. Simplified 62.5%

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

Alternative 14: 57.5% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in w around 0 54.1%

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

Reproduce

herbie shell --seed 2024140 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))