exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 14.7s

Alternatives: 14

Speedup: N/A×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 97.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{1}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (/ 1.0 (exp w))))

C

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

Fortran

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

Java

public static double code(double w, double l) {
	return l * (1.0 / Math.exp(w));
}

Python

def code(w, l):
	return l * (1.0 / math.exp(w))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 98.3%

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

   1. frac-2neg 98.3%

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

   2. div-inv 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

   \[\leadsto \ell \cdot \frac{1}{e^{w}} \]
9. Add Preprocessing

Alternative 3: 97.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 98.3%

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

Alternative 4: 88.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.215:\\ \;\;\;\;\ell - \ell \cdot \left(w \cdot \left(1 - w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.215)
   (- l (* l (* w (- 1.0 (* w (+ 0.5 (* w -0.16666666666666666)))))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w (* w 0.16666666666666666))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.215) {
		tmp = l - (l * (w * (1.0 - (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = l / (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 <= (-0.215d0)) then
        tmp = l - (l * (w * (1.0d0 - (w * (0.5d0 + (w * (-0.16666666666666666d0)))))))
    else
        tmp = l / (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 <= -0.215) {
		tmp = l - (l * (w * (1.0 - (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.215:
		tmp = l - (l * (w * (1.0 - (w * (0.5 + (w * -0.16666666666666666))))))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.215)
		tmp = Float64(l - Float64(l * Float64(w * Float64(1.0 - Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))));
	else
		tmp = Float64(l / 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 <= -0.215)
		tmp = l - (l * (w * (1.0 - (w * (0.5 + (w * -0.16666666666666666))))));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.215], N[(l - N[(l * N[(w * N[(1.0 - N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / 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 < -0.214999999999999997

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 68.2%

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

   7. Taylor expanded in l around 0 77.4%

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

   if -0.214999999999999997 < w

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 97.5%

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

   6. Taylor expanded in w around 0 92.1%

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

      1. *-commutative 92.1%

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

   8. Simplified 92.1%

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

   9. Taylor expanded in w around inf 92.1%

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

       1. *-commutative 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.215:\\ \;\;\;\;\ell - \ell \cdot \left(w \cdot \left(1 - w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.88:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(w \cdot \left(\ell \cdot -0.16666666666666666\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.88)
   (+ l (* w (- (* w (* w (* l -0.16666666666666666))) l)))
   (/ l (+ 1.0 (* w (+ 1.0 (* w (* w 0.16666666666666666))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.88) {
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	} else {
		tmp = l / (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 <= 0.88d0) then
        tmp = l + (w * ((w * (w * (l * (-0.16666666666666666d0)))) - l))
    else
        tmp = l / (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 <= 0.88) {
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.88:
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.88)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(w * Float64(l * -0.16666666666666666))) - l)));
	else
		tmp = Float64(l / 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 <= 0.88)
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * (w * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.88], N[(l + N[(w * N[(N[(w * N[(w * N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / 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 < 0.880000000000000004

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 98.5%

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

   6. Taylor expanded in w around 0 86.6%

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

   7. Taylor expanded in w around inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-rgt-neg-in 86.6%

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

      3. distribute-rgt-out 86.6%

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

      4. metadata-eval 86.6%

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

      5. distribute-neg-in 86.6%

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

      6. mul-1-neg 86.6%

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

      7. distribute-rgt-out 86.6%

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

      8. metadata-eval 86.6%

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

      9. associate-*r* 86.6%

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

      10. metadata-eval 86.6%

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

      11. *-lft-identity 86.6%

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

      12. sub-neg 86.6%

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

      13. distribute-lft-out-- 86.6%

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

      14. metadata-eval 86.6%

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

   9. Simplified 86.6%

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

   if 0.880000000000000004 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 97.5%

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

   6. Taylor expanded in w around 0 72.6%

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

      1. *-commutative 72.6%

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

   8. Simplified 72.6%

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

   9. Taylor expanded in w around inf 72.6%

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

       1. *-commutative 72.6%

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

   11. Simplified 72.6%

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

Alternative 6: 85.2% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(w \cdot \left(\ell \cdot -0.16666666666666666\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2e-11)
   (+ l (* w (- (* w (* w (* l -0.16666666666666666))) l)))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2e-11) {
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	} 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 <= (-2d-11)) then
        tmp = l + (w * ((w * (w * (l * (-0.16666666666666666d0)))) - l))
    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 <= -2e-11) {
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2e-11:
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2e-11)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(w * Float64(l * -0.16666666666666666))) - l)));
	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 <= -2e-11)
		tmp = l + (w * ((w * (w * (l * -0.16666666666666666))) - l));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2e-11], N[(l + N[(w * N[(N[(w * N[(w * N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.99999999999999988e-11

   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 99.4%

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

   6. Taylor expanded in w around 0 68.0%

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

   7. Taylor expanded in w around inf 68.0%

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

      1. mul-1-neg 68.0%

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

      2. distribute-rgt-neg-in 68.0%

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

      3. distribute-rgt-out 68.0%

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

      4. metadata-eval 68.0%

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

      5. distribute-neg-in 68.0%

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

      6. mul-1-neg 68.0%

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

      7. distribute-rgt-out 68.0%

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

      8. metadata-eval 68.0%

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

      9. associate-*r* 68.0%

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

      10. metadata-eval 68.0%

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

      11. *-lft-identity 68.0%

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

      12. sub-neg 68.0%

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

      13. distribute-lft-out-- 68.0%

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

      14. metadata-eval 68.0%

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

   9. Simplified 68.0%

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

   if -1.99999999999999988e-11 < w

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 97.8%

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

   6. Taylor expanded in w around 0 92.3%

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

      1. *-commutative 92.3%

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

   8. Simplified 92.3%

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

Alternative 7: 84.0% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 128:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 - w \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 128.0)
   (* l (+ 1.0 (* w (- -1.0 (* w -0.5)))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 128.0) {
		tmp = l * (1.0 + (w * (-1.0 - (w * -0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 128.0d0) then
        tmp = l * (1.0d0 + (w * ((-1.0d0) - (w * (-0.5d0)))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 128.0) {
		tmp = l * (1.0 + (w * (-1.0 - (w * -0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 128.0:
		tmp = l * (1.0 + (w * (-1.0 - (w * -0.5))))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 128.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 - Float64(w * -0.5)))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 128.0)
		tmp = l * (1.0 + (w * (-1.0 - (w * -0.5))));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 128.0], N[(l * N[(1.0 + N[(w * N[(-1.0 - N[(w * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 128

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 98.0%

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

      1. frac-2neg 98.0%

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

      2. div-inv 98.0%

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

   7. Applied egg-rr 98.0%

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

   8. Taylor expanded in w around 0 85.7%

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

   if 128 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 74.3%

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

      1. *-commutative 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 128:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 - w \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 520:\\ \;\;\;\;\ell - w \cdot \left(\ell - w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 520.0)
   (- l (* w (- l (* w (* l 0.5)))))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 520.0) {
		tmp = l - (w * (l - (w * (l * 0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 520.0d0) then
        tmp = l - (w * (l - (w * (l * 0.5d0))))
    else
        tmp = l / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 520.0) {
		tmp = l - (w * (l - (w * (l * 0.5))));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 520.0:
		tmp = l - (w * (l - (w * (l * 0.5))))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 520.0)
		tmp = Float64(l - Float64(w * Float64(l - Float64(w * Float64(l * 0.5)))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 520.0)
		tmp = l - (w * (l - (w * (l * 0.5))));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 520.0], N[(l - N[(w * N[(l - N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 520

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 98.0%

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

   6. Taylor expanded in w around 0 86.2%

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

   7. Taylor expanded in w around 0 81.0%

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

      1. distribute-rgt-out 81.0%

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

      2. metadata-eval 81.0%

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

      3. neg-mul-1 81.0%

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

      4. associate-*r* 81.0%

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

      5. *-commutative 81.0%

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

      6. distribute-rgt-neg-in 81.0%

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

      7. *-commutative 81.0%

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

      8. metadata-eval 81.0%

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

      9. associate-*r* 81.0%

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

      10. *-commutative 81.0%

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

   9. Simplified 81.0%

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

   if 520 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 74.3%

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

      1. *-commutative 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 520:\\ \;\;\;\;\ell - w \cdot \left(\ell - w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 98:\\ \;\;\;\;\ell - w \cdot \left(\ell - w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 98.0) (- l (* w (- l (* w (* l 0.5))))) (/ 1.0 (/ (+ w 1.0) l))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 98.0) {
		tmp = l - (w * (l - (w * (l * 0.5))));
	} else {
		tmp = 1.0 / ((w + 1.0) / 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 <= 98.0d0) then
        tmp = l - (w * (l - (w * (l * 0.5d0))))
    else
        tmp = 1.0d0 / ((w + 1.0d0) / l)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 98.0) {
		tmp = l - (w * (l - (w * (l * 0.5))));
	} else {
		tmp = 1.0 / ((w + 1.0) / l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 98.0:
		tmp = l - (w * (l - (w * (l * 0.5))))
	else:
		tmp = 1.0 / ((w + 1.0) / l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 98.0)
		tmp = Float64(l - Float64(w * Float64(l - Float64(w * Float64(l * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(Float64(w + 1.0) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 98.0)
		tmp = l - (w * (l - (w * (l * 0.5))));
	else
		tmp = 1.0 / ((w + 1.0) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 98.0], N[(l - N[(w * N[(l - N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(w + 1.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 98

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 98.0%

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

   6. Taylor expanded in w around 0 86.2%

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

   7. Taylor expanded in w around 0 81.0%

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

      1. distribute-rgt-out 81.0%

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

      2. metadata-eval 81.0%

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

      3. neg-mul-1 81.0%

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

      4. associate-*r* 81.0%

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

      5. *-commutative 81.0%

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

      6. distribute-rgt-neg-in 81.0%

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

      7. *-commutative 81.0%

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

      8. metadata-eval 81.0%

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

      9. associate-*r* 81.0%

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

      10. *-commutative 81.0%

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

   9. Simplified 81.0%

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

   if 98 < w

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. pow1 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. sqr-neg 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. pow1 100.0%

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

      13. inv-pow 100.0%

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

      14. exp-neg 100.0%

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

      15. pow1 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. exp-neg 100.0%

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

      19. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-div 100.0%

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

      3. associate-/r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 49.0%

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

      1. +-commutative 49.0%

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

   9. Simplified 49.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 98:\\ \;\;\;\;\ell - w \cdot \left(\ell - w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.7% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.125:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.125) (* l (- 1.0 w)) (/ 1.0 (/ (+ w 1.0) l))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.125) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 1.0 / ((w + 1.0) / l);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.125d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = 1.0d0 / ((w + 1.0d0) / l)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.125) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 1.0 / ((w + 1.0) / l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.125:
		tmp = l * (1.0 - w)
	else:
		tmp = 1.0 / ((w + 1.0) / l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.125)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(1.0 / Float64(Float64(w + 1.0) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.125)
		tmp = l * (1.0 - w);
	else
		tmp = 1.0 / ((w + 1.0) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.125], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(w + 1.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.125

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0 64.2%

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

      1. neg-mul-1 64.2%

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

      2. unsub-neg 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in w around 0 72.5%

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

   if 0.125 < w

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 2.6%

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

      6. pow1 2.6%

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

      7. add-sqr-sqrt 2.6%

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

      8. sqrt-unprod 2.6%

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

      9. sqr-neg 2.6%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. pow1 100.0%

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

      13. inv-pow 100.0%

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

      14. exp-neg 100.0%

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

      15. pow1 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. exp-neg 100.0%

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

      19. associate-/r/ 100.0%

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

   4. Applied egg-rr 97.5%

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

      1. exp-neg 97.5%

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

      2. remove-double-div 97.5%

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

      3. associate-/r/ 97.5%

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in w around 0 47.8%

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

      1. +-commutative 47.8%

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

   9. Simplified 47.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.125:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.13:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.13) (* l (- 1.0 w)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l * (1.0 - w);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.13d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l * (1.0 - w);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.13:
		tmp = l * (1.0 - w)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.13)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.13)
		tmp = l * (1.0 - w);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.13], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.13

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0 64.2%

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

      1. neg-mul-1 64.2%

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

      2. unsub-neg 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in w around 0 72.5%

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

   if 0.13 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 97.5%

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

   6. Taylor expanded in w around 0 44.6%

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

      1. +-commutative 47.8%

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

   8. Simplified 44.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.13:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.021:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.021) (* l (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.021) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.021d0)) then
        tmp = l * -w
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.021) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.021:
		tmp = l * -w
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.021)
		tmp = Float64(l * Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.021)
		tmp = l * -w;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.021], N[(l * (-w)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.0210000000000000013

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0 5.4%

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

      1. neg-mul-1 5.4%

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

      2. unsub-neg 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in w around 0 30.1%

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

   7. Taylor expanded in w around inf 30.1%

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

      1. neg-mul-1 30.1%

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

   9. Simplified 30.1%

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

   if -0.0210000000000000013 < w

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 97.5%

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

   6. Taylor expanded in w around 0 77.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.021:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.5% 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 99.8%

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

3. Taylor expanded in w around 0 69.1%

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

   1. neg-mul-1 69.1%

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

   2. unsub-neg 69.1%

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

5. Simplified 69.1%

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

6. Taylor expanded in w around 0 62.2%

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

7. Final simplification 62.2%

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

Alternative 14: 57.2% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 98.3%

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

6. Taylor expanded in w around 0 54.0%

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

Reproduce

herbie shell --seed 2024114 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))