exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 22.1s

Alternatives: 18

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 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.6d0)) then
        tmp = l / exp(w)
    else
        tmp = (l ** exp(w)) / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. 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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 97.4%

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

      17. metadata-eval 97.4%

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

      18. metadata-eval 97.4%

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

      19. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in w around inf 97.4%

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

      1. exp-neg 97.4%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

   7. Simplified 97.4%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

Alternative 3: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666))))))))
   (if (<= w -1.6) (/ l (exp w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.6) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0)))))
    if (w <= (-1.6d0)) then
        tmp = l / exp(w)
    else
        tmp = (l ** t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.6) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, t_0) / t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))))
	tmp = 0
	if w <= -1.6:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, t_0) / t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))))
	tmp = 0.0
	if (w <= -1.6)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	tmp = 0.0;
	if (w <= -1.6)
		tmp = l / exp(w);
	else
		tmp = (l ^ t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.6], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   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 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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 97.4%

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

      17. metadata-eval 97.4%

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

      18. metadata-eval 97.4%

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

      19. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in w around inf 97.4%

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

      1. exp-neg 97.4%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

   7. Simplified 97.4%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in w around 0 98.9%

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

      1. *-commutative 99.2%

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

   10. Simplified 98.9%

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

Alternative 4: 99.3% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.4)
   (/ l (exp w))
   (/
    (pow l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))
    (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.4) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -2.4) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.4:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.4)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))))) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -2.4)
		tmp = l / exp(w);
	else
		tmp = (l ^ (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.4], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -2.39999999999999991

   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 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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 97.4%

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

      17. metadata-eval 97.4%

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

      18. metadata-eval 97.4%

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

      19. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in w around inf 97.4%

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

      1. exp-neg 97.4%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

   7. Simplified 97.4%

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

   if -2.39999999999999991 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in w around 0 98.8%

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

      1. *-commutative 99.2%

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

   10. Simplified 98.8%

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

Alternative 5: 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 99.6%

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

   1. add-sqr-sqrt 42.8%

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

   2. sqrt-unprod 83.7%

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

   3. sqr-neg 83.7%

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

   4. sqrt-unprod 40.9%

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

   5. add-sqr-sqrt 82.3%

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

   6. add-sqr-sqrt 82.3%

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

   7. sqrt-unprod 82.3%

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

   8. add-sqr-sqrt 40.9%

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

   9. sqrt-unprod 70.3%

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

   10. sqr-neg 70.3%

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

   11. sqrt-unprod 29.4%

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

   12. add-sqr-sqrt 56.6%

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

   13. pow1 56.6%

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

   14. exp-neg 56.6%

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

   15. inv-pow 56.6%

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

   16. pow-prod-up 96.9%

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

   17. metadata-eval 96.9%

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

   18. metadata-eval 96.9%

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

   19. metadata-eval 96.9%

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

4. Applied egg-rr 96.9%

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

5. Taylor expanded in w around inf 96.9%

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

Alternative 6: 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 99.6%

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

   1. add-sqr-sqrt 42.8%

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

   2. sqrt-unprod 83.7%

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

   3. sqr-neg 83.7%

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

   4. sqrt-unprod 40.9%

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

   5. add-sqr-sqrt 82.3%

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

   6. add-sqr-sqrt 82.3%

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

   7. sqrt-unprod 82.3%

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

   8. add-sqr-sqrt 40.9%

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

   9. sqrt-unprod 70.3%

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

   10. sqr-neg 70.3%

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

   11. sqrt-unprod 29.4%

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

   12. add-sqr-sqrt 56.6%

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

   13. pow1 56.6%

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

   14. exp-neg 56.6%

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

   15. inv-pow 56.6%

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

   16. pow-prod-up 96.9%

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

   17. metadata-eval 96.9%

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

   18. metadata-eval 96.9%

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

   19. metadata-eval 96.9%

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

4. Applied egg-rr 96.9%

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

5. Taylor expanded in w around inf 96.9%

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

   1. exp-neg 96.9%

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

   2. associate-*r/ 96.9%

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

   3. *-rgt-identity 96.9%

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

7. Simplified 96.9%

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

Alternative 7: 90.9% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.04:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + 1\right) + -1}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.04)
   (* l (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (/
    (+ (+ l 1.0) -1.0)
    (+ 1.0 (* w (+ 1.0 (* w (* w 0.16666666666666666))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.04) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = ((l + 1.0) + -1.0) / (1.0 + (w * (1.0 + (w * (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 <= 0.04d0) then
        tmp = l * (1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = ((l + 1.0d0) + (-1.0d0)) / (1.0d0 + (w * (1.0d0 + (w * (w * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.04) {
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = ((l + 1.0) + -1.0) / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.04:
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = ((l + 1.0) + -1.0) / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.04)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(Float64(Float64(l + 1.0) + -1.0) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(w * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.04)
		tmp = l * (1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = ((l + 1.0) + -1.0) / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.04], 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[(N[(N[(l + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.0400000000000000008

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 81.2%

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

      3. sqr-neg 81.2%

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

      4. sqrt-unprod 46.9%

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

      5. add-sqr-sqrt 80.5%

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

      6. add-sqr-sqrt 80.5%

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

      7. sqrt-unprod 80.5%

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

      8. add-sqr-sqrt 46.9%

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

      9. sqrt-unprod 80.6%

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

      10. sqr-neg 80.6%

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

      11. sqrt-unprod 33.6%

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

      12. add-sqr-sqrt 64.9%

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

      13. pow1 64.9%

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

      14. exp-neg 64.9%

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

      15. inv-pow 64.9%

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

      16. pow-prod-up 97.2%

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

      17. metadata-eval 97.2%

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

      18. metadata-eval 97.2%

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

      19. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

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

   5. Taylor expanded in w around inf 97.2%

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

   6. Taylor expanded in w around 0 91.6%

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

   if 0.0400000000000000008 < w

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. sqr-neg 99.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 94.3%

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

      6. add-sqr-sqrt 94.3%

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

      7. sqrt-unprod 94.3%

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

      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 0.7%

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

      10. sqr-neg 0.7%

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

      11. sqrt-unprod 0.7%

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

      12. add-sqr-sqrt 0.7%

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

      13. pow1 0.7%

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

      14. exp-neg 0.7%

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

      15. inv-pow 0.7%

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

      16. pow-prod-up 94.6%

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

      17. metadata-eval 94.6%

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

      18. metadata-eval 94.6%

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

      19. metadata-eval 94.6%

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

   9. Applied egg-rr 72.8%

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

   10. Taylor expanded in w around inf 72.8%

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

       1. *-commutative 72.8%

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

   12. Simplified 72.8%

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

       1. *-rgt-identity 72.8%

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

       2. expm1-log1p-u 72.8%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell\right)\right)}}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)} \]

       3. expm1-undefine 91.6%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1}}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)} \]

       4. log1p-undefine 91.6%

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

       5. rem-exp-log 91.6%

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

       6. +-commutative 91.6%

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

   14. Applied egg-rr 91.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.04:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + 1\right) + -1}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -85:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 - w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -85.0)
   (* 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 <= -85.0) {
		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 <= (-85.0d0)) 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 <= -85.0) {
		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 <= -85.0:
		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 <= -85.0)
		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 <= -85.0)
		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, -85.0], 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 < -85

   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 48.0%

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

      3. sqr-neg 48.0%

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

      4. sqrt-unprod 48.0%

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

      5. add-sqr-sqrt 48.0%

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

      6. add-sqr-sqrt 48.0%

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

      7. sqrt-unprod 48.0%

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

      8. add-sqr-sqrt 48.0%

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

      9. sqrt-unprod 48.0%

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

      10. sqr-neg 48.0%

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

      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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.7%

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

      17. metadata-eval 98.7%

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

      18. metadata-eval 98.7%

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

      19. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in w around inf 98.7%

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

   6. Taylor expanded in w around 0 81.5%

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

   if -85 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

      1. add-sqr-sqrt 59.8%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 38.1%

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

      5. add-sqr-sqrt 96.0%

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

      6. add-sqr-sqrt 96.0%

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

      7. sqrt-unprod 96.0%

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

      8. add-sqr-sqrt 38.1%

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

      9. sqrt-unprod 79.1%

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

      10. sqr-neg 79.1%

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

      11. sqrt-unprod 41.1%

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

      12. add-sqr-sqrt 79.2%

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

      13. pow1 79.2%

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

      14. exp-neg 79.2%

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

      15. inv-pow 79.2%

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

      16. pow-prod-up 96.1%

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

      17. metadata-eval 96.1%

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

      18. metadata-eval 96.1%

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

      19. metadata-eval 96.1%

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

   9. Applied egg-rr 92.2%

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

       1. add-sqr-sqrt 54.1%

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

       2. sqrt-unprod 92.2%

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

       3. sqr-neg 92.2%

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

       4. sqrt-unprod 38.1%

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

       5. add-sqr-sqrt 92.2%

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

       6. cancel-sign-sub-inv 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -85:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 - w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 8e-9)
   (* 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 <= 8e-9) {
		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 <= 8d-9) 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 <= 8e-9) {
		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 <= 8e-9:
		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 <= 8e-9)
		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 <= 8e-9)
		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, 8e-9], 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 < 8.0000000000000005e-9

   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 33.8%

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

      2. sqrt-unprod 81.2%

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

      3. sqr-neg 81.2%

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

      4. sqrt-unprod 47.4%

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

      5. add-sqr-sqrt 81.1%

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

      6. add-sqr-sqrt 81.1%

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

      7. sqrt-unprod 81.1%

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

      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 81.1%

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

      10. sqr-neg 81.1%

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

      11. sqrt-unprod 33.7%

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

      12. add-sqr-sqrt 65.3%

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

      13. pow1 65.3%

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

      14. exp-neg 65.3%

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

      15. inv-pow 65.3%

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

      16. pow-prod-up 97.9%

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

      17. metadata-eval 97.9%

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

      18. metadata-eval 97.9%

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

      19. metadata-eval 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in w around inf 97.9%

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

   6. Taylor expanded in w around 0 92.2%

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

   if 8.0000000000000005e-9 < w

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

      1. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 99.2%

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

      3. sqr-neg 99.2%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 90.1%

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

      6. add-sqr-sqrt 90.1%

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

      7. sqrt-unprod 90.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.8%

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

      10. sqr-neg 1.8%

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

      11. sqrt-unprod 1.8%

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

      12. add-sqr-sqrt 1.8%

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

      13. pow1 1.8%

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

      14. exp-neg 1.8%

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

      15. inv-pow 1.8%

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

      16. pow-prod-up 90.4%

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

      17. metadata-eval 90.4%

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

      18. metadata-eval 90.4%

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

      19. metadata-eval 90.4%

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

   9. Applied egg-rr 69.8%

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

   10. Taylor expanded in l around 0 69.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -15.2:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -15.2)
   (* l (+ 1.0 (* 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 <= -15.2) {
		tmp = l * (1.0 + (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 <= (-15.2d0)) 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))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -15.2) {
		tmp = l * (1.0 + (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 <= -15.2:
		tmp = l * (1.0 + (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 <= -15.2)
		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 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -15.2)
		tmp = l * (1.0 + (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, -15.2], 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 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -15.199999999999999

   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 48.0%

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

      3. sqr-neg 48.0%

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

      4. sqrt-unprod 48.0%

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

      5. add-sqr-sqrt 48.0%

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

      6. add-sqr-sqrt 48.0%

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

      7. sqrt-unprod 48.0%

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

      8. add-sqr-sqrt 48.0%

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

      9. sqrt-unprod 48.0%

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

      10. sqr-neg 48.0%

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

      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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.7%

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

      17. metadata-eval 98.7%

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

      18. metadata-eval 98.7%

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

      19. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in w around inf 98.7%

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

   6. Taylor expanded in w around 0 81.5%

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

   if -15.199999999999999 < w

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 59.8%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 38.1%

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

      5. add-sqr-sqrt 96.0%

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

      6. add-sqr-sqrt 96.0%

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

      7. sqrt-unprod 96.0%

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

      8. add-sqr-sqrt 38.1%

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

      9. sqrt-unprod 79.1%

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

      10. sqr-neg 79.1%

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

      11. sqrt-unprod 41.1%

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

      12. add-sqr-sqrt 79.2%

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

      13. pow1 79.2%

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

      14. exp-neg 79.2%

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

      15. inv-pow 79.2%

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

      16. pow-prod-up 96.1%

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

      17. metadata-eval 96.1%

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

      18. metadata-eval 96.1%

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

      19. metadata-eval 96.1%

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

   4. Applied egg-rr 96.1%

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

   5. Taylor expanded in w around inf 96.1%

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

      1. exp-neg 96.1%

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

      2. associate-*r/ 96.1%

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

      3. *-rgt-identity 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in w around 0 91.5%

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

      1. *-commutative 98.4%

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

   10. Simplified 91.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -15.2:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -100:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -100.0)
   (* l (+ 1.0 (* w (+ (* w 0.5) -1.0))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -100.0) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -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 <= (-100.0d0)) then
        tmp = l * (1.0d0 + (w * ((w * 0.5d0) + (-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 <= -100.0) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -100.0:
		tmp = l * (1.0 + (w * ((w * 0.5) + -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 <= -100.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -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 <= -100.0)
		tmp = l * (1.0 + (w * ((w * 0.5) + -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, -100.0], N[(l * N[(1.0 + N[(w * N[(N[(w * 0.5), $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 < -100

   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 48.0%

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

      3. sqr-neg 48.0%

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

      4. sqrt-unprod 48.0%

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

      5. add-sqr-sqrt 48.0%

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

      6. add-sqr-sqrt 48.0%

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

      7. sqrt-unprod 48.0%

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

      8. add-sqr-sqrt 48.0%

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

      9. sqrt-unprod 48.0%

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

      10. sqr-neg 48.0%

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

      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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.7%

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

      17. metadata-eval 98.7%

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

      18. metadata-eval 98.7%

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

      19. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in w around inf 98.7%

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

   6. Taylor expanded in w around 0 63.2%

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

   if -100 < w

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 59.8%

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

      2. sqrt-unprod 97.9%

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

      3. sqr-neg 97.9%

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

      4. sqrt-unprod 38.1%

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

      5. add-sqr-sqrt 96.0%

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

      6. add-sqr-sqrt 96.0%

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

      7. sqrt-unprod 96.0%

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

      8. add-sqr-sqrt 38.1%

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

      9. sqrt-unprod 79.1%

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

      10. sqr-neg 79.1%

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

      11. sqrt-unprod 41.1%

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

      12. add-sqr-sqrt 79.2%

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

      13. pow1 79.2%

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

      14. exp-neg 79.2%

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

      15. inv-pow 79.2%

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

      16. pow-prod-up 96.1%

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

      17. metadata-eval 96.1%

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

      18. metadata-eval 96.1%

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

      19. metadata-eval 96.1%

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

   4. Applied egg-rr 96.1%

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

   5. Taylor expanded in w around inf 96.1%

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

      1. exp-neg 96.1%

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

      2. associate-*r/ 96.1%

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

      3. *-rgt-identity 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in w around 0 91.5%

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

      1. *-commutative 98.4%

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

   10. Simplified 91.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -100:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.035:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.035) (* l (+ 1.0 (* w (+ (* w 0.5) -1.0)))) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.035) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.035d0) then
        tmp = l * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.035) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.035:
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.035)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.035)
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.035], N[(l * N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.035000000000000003

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-unprod 81.2%

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

      3. sqr-neg 81.2%

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

      4. sqrt-unprod 46.9%

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

      5. add-sqr-sqrt 80.5%

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

      6. add-sqr-sqrt 80.5%

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

      7. sqrt-unprod 80.5%

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

      8. add-sqr-sqrt 46.9%

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

      9. sqrt-unprod 80.6%

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

      10. sqr-neg 80.6%

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

      11. sqrt-unprod 33.6%

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

      12. add-sqr-sqrt 64.9%

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

      13. pow1 64.9%

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

      14. exp-neg 64.9%

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

      15. inv-pow 64.9%

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

      16. pow-prod-up 97.2%

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

      17. metadata-eval 97.2%

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

      18. metadata-eval 97.2%

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

      19. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

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

   5. Taylor expanded in w around inf 97.2%

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

   6. Taylor expanded in w around 0 85.6%

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

   if 0.035000000000000003 < w

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. sqr-neg 99.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 94.3%

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

      6. add-sqr-sqrt 94.3%

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

      7. sqrt-unprod 94.3%

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

      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 0.7%

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

      10. sqr-neg 0.7%

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

      11. sqrt-unprod 0.7%

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

      12. add-sqr-sqrt 0.7%

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

      13. pow1 0.7%

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

      14. exp-neg 0.7%

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

      15. inv-pow 0.7%

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

      16. pow-prod-up 94.6%

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

      17. metadata-eval 94.6%

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

      18. metadata-eval 94.6%

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

      19. metadata-eval 94.6%

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

   4. Applied egg-rr 94.6%

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

   5. Taylor expanded in w around inf 94.6%

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

      1. exp-neg 94.6%

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

      2. associate-*r/ 94.6%

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

      3. *-rgt-identity 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in w around 0 38.2%

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

      1. +-commutative 38.2%

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

   10. Simplified 38.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.035:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.5% accurate, 21.8× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.122) {
		tmp = l + (w * ((l * w) - l));
	} 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 <= 0.122d0) then
        tmp = l + (w * ((l * w) - l))
    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 <= 0.122) {
		tmp = l + (w * ((l * w) - l));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < 0.122

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 66.8%

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

      1. *-commutative 66.8%

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

   7. Simplified 66.8%

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

      1. add-sqr-sqrt 34.6%

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

      2. sqrt-unprod 81.3%

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

      3. sqr-neg 81.3%

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

      4. sqrt-unprod 46.7%

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

      5. add-sqr-sqrt 80.2%

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

      6. add-sqr-sqrt 80.2%

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

      7. sqrt-unprod 80.2%

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

      8. add-sqr-sqrt 46.7%

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

      9. sqrt-unprod 80.2%

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

      10. sqr-neg 80.2%

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

      11. sqrt-unprod 33.5%

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

      12. add-sqr-sqrt 64.7%

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

      13. pow1 64.7%

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

      14. exp-neg 64.7%

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

      15. inv-pow 64.7%

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

      16. pow-prod-up 96.8%

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

      17. metadata-eval 96.8%

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

      18. metadata-eval 96.8%

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

      19. metadata-eval 96.8%

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

   9. Applied egg-rr 65.1%

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

   10. Taylor expanded in w around inf 65.1%

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

       1. *-commutative 65.1%

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

   12. Simplified 65.1%

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

   13. Taylor expanded in w around 0 82.3%

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

   if 0.122 < 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 97.0%

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

      6. add-sqr-sqrt 97.0%

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

      7. sqrt-unprod 97.0%

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

      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 0.3%

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

      10. sqr-neg 0.3%

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

      11. sqrt-unprod 0.3%

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

      12. add-sqr-sqrt 0.3%

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

      13. pow1 0.3%

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

      14. exp-neg 0.3%

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

      15. inv-pow 0.3%

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

      16. pow-prod-up 97.2%

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

      17. metadata-eval 97.2%

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

      18. metadata-eval 97.2%

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

      19. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

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

   5. Taylor expanded in w around inf 97.2%

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

      1. exp-neg 97.2%

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

      2. associate-*r/ 97.2%

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

      3. *-rgt-identity 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in w around 0 39.0%

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

      1. +-commutative 39.0%

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

   10. Simplified 39.0%

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

Alternative 14: 70.8% accurate, 30.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. 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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 97.4%

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

      17. metadata-eval 97.4%

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

      18. metadata-eval 97.4%

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

      19. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in w around inf 97.4%

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

   6. Taylor expanded in w around 0 28.8%

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

      1. neg-mul-1 28.8%

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

      2. unsub-neg 28.8%

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

   8. Simplified 28.8%

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

   9. Taylor expanded in w around inf 28.8%

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

       1. mul-1-neg 28.8%

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

       2. distribute-rgt-neg-out 28.8%

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

   11. Simplified 28.8%

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

   if -1 < w

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 60.2%

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

      2. sqrt-unprod 98.4%

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

      3. sqr-neg 98.4%

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

      4. sqrt-unprod 38.2%

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

      5. add-sqr-sqrt 96.5%

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

      6. add-sqr-sqrt 96.5%

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

      7. sqrt-unprod 96.5%

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

      8. add-sqr-sqrt 38.2%

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

      9. sqrt-unprod 79.6%

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

      10. sqr-neg 79.6%

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

      11. sqrt-unprod 41.3%

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

      12. add-sqr-sqrt 79.6%

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

      13. pow1 79.6%

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

      14. exp-neg 79.6%

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

      15. inv-pow 79.6%

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

      16. pow-prod-up 96.6%

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

      17. metadata-eval 96.6%

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

      18. metadata-eval 96.6%

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

      19. metadata-eval 96.6%

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in w around inf 96.6%

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

      1. exp-neg 96.6%

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

      2. associate-*r/ 96.6%

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

      3. *-rgt-identity 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in w around 0 86.4%

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

      1. +-commutative 86.4%

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

   10. Simplified 86.4%

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

4. Final simplification 69.8%

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

Alternative 15: 64.2% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -15.5:\\ \;\;\;\;-\ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -15.5) (- (* l w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -15.5) {
		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 <= (-15.5d0)) 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 <= -15.5) {
		tmp = -(l * w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -15.5:
		tmp = -(l * w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -15.5)
		tmp = Float64(-Float64(l * w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -15.5)
		tmp = -(l * w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -15.5], (-N[(l * w), $MachinePrecision]), l]

Derivation

1. Split input into 2 regimes

2. if w < -15.5

   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 48.0%

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

      3. sqr-neg 48.0%

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

      4. sqrt-unprod 48.0%

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

      5. add-sqr-sqrt 48.0%

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

      6. add-sqr-sqrt 48.0%

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

      7. sqrt-unprod 48.0%

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

      8. add-sqr-sqrt 48.0%

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

      9. sqrt-unprod 48.0%

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

      10. sqr-neg 48.0%

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

      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 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.7%

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

      17. metadata-eval 98.7%

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

      18. metadata-eval 98.7%

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

      19. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in w around inf 98.7%

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

   6. Taylor expanded in w around 0 29.2%

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

      1. neg-mul-1 29.2%

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

      2. unsub-neg 29.2%

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

   8. Simplified 29.2%

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

   9. Taylor expanded in w around inf 29.2%

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

       1. mul-1-neg 29.2%

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

       2. distribute-rgt-neg-out 29.2%

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

   11. Simplified 29.2%

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

   if -15.5 < w

   1. Initial program 99.4%

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

   3. Taylor expanded in w around 0 80.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -15.5:\\ \;\;\;\;-\ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.9% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 42.8%

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

   2. sqrt-unprod 83.7%

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

   3. sqr-neg 83.7%

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

   4. sqrt-unprod 40.9%

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

   5. add-sqr-sqrt 82.3%

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

   6. add-sqr-sqrt 82.3%

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

   7. sqrt-unprod 82.3%

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

   8. add-sqr-sqrt 40.9%

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

   9. sqrt-unprod 70.3%

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

   10. sqr-neg 70.3%

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

   11. sqrt-unprod 29.4%

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

   12. add-sqr-sqrt 56.6%

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

   13. pow1 56.6%

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

   14. exp-neg 56.6%

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

   15. inv-pow 56.6%

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

   16. pow-prod-up 96.9%

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

   17. metadata-eval 96.9%

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

   18. metadata-eval 96.9%

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

   19. metadata-eval 96.9%

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

4. Applied egg-rr 96.9%

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

5. Taylor expanded in w around inf 96.9%

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

6. Taylor expanded in w around 0 65.3%

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

   1. mul-1-neg 65.3%

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

   2. unsub-neg 65.3%

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

8. Simplified 65.3%

   \[\leadsto \color{blue}{\ell - \ell \cdot w} \]
9. Add Preprocessing

Alternative 17: 63.9% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 42.8%

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

   2. sqrt-unprod 83.7%

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

   3. sqr-neg 83.7%

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

   4. sqrt-unprod 40.9%

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

   5. add-sqr-sqrt 82.3%

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

   6. add-sqr-sqrt 82.3%

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

   7. sqrt-unprod 82.3%

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

   8. add-sqr-sqrt 40.9%

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

   9. sqrt-unprod 70.3%

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

   10. sqr-neg 70.3%

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

   11. sqrt-unprod 29.4%

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

   12. add-sqr-sqrt 56.6%

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

   13. pow1 56.6%

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

   14. exp-neg 56.6%

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

   15. inv-pow 56.6%

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

   16. pow-prod-up 96.9%

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

   17. metadata-eval 96.9%

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

   18. metadata-eval 96.9%

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

   19. metadata-eval 96.9%

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

4. Applied egg-rr 96.9%

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

5. Taylor expanded in w around inf 96.9%

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

6. Taylor expanded in w around 0 65.3%

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

   1. neg-mul-1 65.3%

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

   2. unsub-neg 65.3%

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

8. Simplified 65.3%

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

Alternative 18: 57.7% 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 99.6%

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

3. Taylor expanded in w around 0 58.3%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))