exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 23.6s

Alternatives: 7

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

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)}}{\frac{e^{w + 1}}{e}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (/ (exp (+ w 1.0)) E)))

C

double code(double w, double l) {
	return pow(l, exp(w)) / (exp((w + 1.0)) / ((double) M_E));
}

Java

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

Python

def code(w, l):
	return math.pow(l, math.exp(w)) / (math.exp((w + 1.0)) / math.e)

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / Float64(exp(Float64(w + 1.0)) / exp(1)))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / (exp((w + 1.0)) / 2.71828182845904523536);
end

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[Exp[N[(w + 1.0), $MachinePrecision]], $MachinePrecision] / E), $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. Step-by-step derivation

   1. expm1-log1p-u 74.1%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

   2. expm1-undefine 74.1%

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

   3. exp-diff 74.1%

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

   4. log1p-undefine 74.1%

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

   5. rem-exp-log 99.8%

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

   6. exp-1-e 99.8%

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

6. Applied egg-rr 99.8%

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

   1. +-commutative 99.8%

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

8. Simplified 99.8%

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

Alternative 2: 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 3: 97.5% 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. Step-by-step derivation

   1. expm1-log1p-u 74.1%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

   2. expm1-undefine 74.1%

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

   3. exp-diff 74.1%

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

   4. log1p-undefine 74.1%

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

   5. rem-exp-log 99.8%

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

   6. exp-1-e 99.8%

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

6. Applied egg-rr 99.8%

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

   1. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in l around 0 99.4%

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

10. Taylor expanded in w around 0 97.9%

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

    1. *-commutative 97.9%

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

12. Simplified 97.9%

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

13. Taylor expanded in l around 0 97.9%

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

    1. +-commutative 97.9%

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

    2. exp-sum 97.9%

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

    3. exp-1-e 97.9%

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

    4. associate-/l/ 97.9%

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

    5. associate-/l* 98.4%

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

    6. *-inverses 98.4%

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

    7. *-rgt-identity 98.4%

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

15. Simplified 98.4%

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

Alternative 4: 73.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot e}{e + w \cdot \left(e + 0.5 \cdot \left(w \cdot e\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 3.8e-29)
   (- l (* l w))
   (/ (* l E) (+ E (* w (+ E (* 0.5 (* w E))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 3.8e-29) {
		tmp = l - (l * w);
	} else {
		tmp = (l * ((double) M_E)) / (((double) M_E) + (w * (((double) M_E) + (0.5 * (w * ((double) M_E))))));
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 3.8e-29) {
		tmp = l - (l * w);
	} else {
		tmp = (l * Math.E) / (Math.E + (w * (Math.E + (0.5 * (w * Math.E)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 3.8e-29:
		tmp = l - (l * w)
	else:
		tmp = (l * math.e) / (math.e + (w * (math.e + (0.5 * (w * math.e)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 3.8e-29)
		tmp = Float64(l - Float64(l * w));
	else
		tmp = Float64(Float64(l * exp(1)) / Float64(exp(1) + Float64(w * Float64(exp(1) + Float64(0.5 * Float64(w * exp(1)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 3.8e-29)
		tmp = l - (l * w);
	else
		tmp = (l * 2.71828182845904523536) / (2.71828182845904523536 + (w * (2.71828182845904523536 + (0.5 * (w * 2.71828182845904523536)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 3.8e-29], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision], N[(N[(l * E), $MachinePrecision] / N[(E + N[(w * N[(E + N[(0.5 * N[(w * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.79999999999999976e-29

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 69.3%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

      2. expm1-undefine 69.3%

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

      3. exp-diff 69.3%

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

      4. log1p-undefine 69.3%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in l around 0 99.4%

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

   10. Taylor expanded in w around 0 98.3%

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

       1. *-commutative 98.3%

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

   12. Simplified 98.3%

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

   13. Taylor expanded in w around 0 76.6%

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

       1. +-commutative 76.6%

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

       2. mul-1-neg 76.6%

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

       3. unsub-neg 76.6%

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

       4. exp-1-e 76.6%

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

       5. associate-/l* 77.2%

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

       6. *-inverses 77.2%

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

       7. *-rgt-identity 77.2%

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

       8. exp-1-e 77.2%

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

       9. associate-/l* 77.2%

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

       10. associate-/l* 77.2%

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

       11. *-inverses 77.2%

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

       12. *-rgt-identity 77.2%

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

       13. *-commutative 77.2%

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

   15. Simplified 77.2%

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

   if 3.79999999999999976e-29 < 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. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

      2. expm1-undefine 99.2%

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

      3. exp-diff 99.2%

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

      4. log1p-undefine 99.2%

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

      5. rem-exp-log 99.2%

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

      6. exp-1-e 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in l around 0 99.2%

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

   10. Taylor expanded in w around 0 95.4%

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

       1. *-commutative 95.4%

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

   12. Simplified 95.4%

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

   13. Taylor expanded in w around 0 78.3%

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

       1. exp-1-e 78.3%

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

       2. exp-1-e 78.3%

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

       3. exp-1-e 78.3%

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

       4. *-commutative 78.3%

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

   15. Simplified 78.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot e}{e + w \cdot \left(e + 0.5 \cdot \left(w \cdot e\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot e}{\left(w + 1\right) \cdot e}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 3.8e-29) (- l (* l w)) (/ (* l E) (* (+ w 1.0) E))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 3.8e-29) {
		tmp = l - (l * w);
	} else {
		tmp = (l * ((double) M_E)) / ((w + 1.0) * ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 3.8e-29) {
		tmp = l - (l * w);
	} else {
		tmp = (l * Math.E) / ((w + 1.0) * Math.E);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 3.8e-29:
		tmp = l - (l * w)
	else:
		tmp = (l * math.e) / ((w + 1.0) * math.e)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 3.8e-29)
		tmp = Float64(l - Float64(l * w));
	else
		tmp = Float64(Float64(l * exp(1)) / Float64(Float64(w + 1.0) * exp(1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 3.8e-29)
		tmp = l - (l * w);
	else
		tmp = (l * 2.71828182845904523536) / ((w + 1.0) * 2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 3.8e-29], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision], N[(N[(l * E), $MachinePrecision] / N[(N[(w + 1.0), $MachinePrecision] * E), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.79999999999999976e-29

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 69.3%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

      2. expm1-undefine 69.3%

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

      3. exp-diff 69.3%

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

      4. log1p-undefine 69.3%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in l around 0 99.4%

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

   10. Taylor expanded in w around 0 98.3%

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

       1. *-commutative 98.3%

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

   12. Simplified 98.3%

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

   13. Taylor expanded in w around 0 76.6%

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

       1. +-commutative 76.6%

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

       2. mul-1-neg 76.6%

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

       3. unsub-neg 76.6%

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

       4. exp-1-e 76.6%

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

       5. associate-/l* 77.2%

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

       6. *-inverses 77.2%

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

       7. *-rgt-identity 77.2%

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

       8. exp-1-e 77.2%

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

       9. associate-/l* 77.2%

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

       10. associate-/l* 77.2%

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

       11. *-inverses 77.2%

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

       12. *-rgt-identity 77.2%

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

       13. *-commutative 77.2%

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

   15. Simplified 77.2%

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

   if 3.79999999999999976e-29 < 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. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

      2. expm1-undefine 99.2%

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

      3. exp-diff 99.2%

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

      4. log1p-undefine 99.2%

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

      5. rem-exp-log 99.2%

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

      6. exp-1-e 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in l around 0 99.2%

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

   10. Taylor expanded in w around 0 95.4%

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

       1. *-commutative 95.4%

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

   12. Simplified 95.4%

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

   13. Taylor expanded in w around 0 52.4%

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

       1. exp-1-e 52.4%

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

       2. exp-1-e 52.4%

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

       3. distribute-rgt1-in 52.4%

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

   15. Simplified 52.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot e}{\left(w + 1\right) \cdot e}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 99.8%

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

   1. expm1-log1p-u 74.1%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

   2. expm1-undefine 74.1%

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

   3. exp-diff 74.1%

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

   4. log1p-undefine 74.1%

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

   5. rem-exp-log 99.8%

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

   6. exp-1-e 99.8%

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

6. Applied egg-rr 99.8%

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

   1. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in l around 0 99.4%

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

10. Taylor expanded in w around 0 97.9%

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

    1. *-commutative 97.9%

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

12. Simplified 97.9%

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

13. Taylor expanded in w around 0 66.7%

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

    1. +-commutative 66.7%

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

    2. mul-1-neg 66.7%

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

    3. unsub-neg 66.7%

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

    4. exp-1-e 66.7%

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

    5. associate-/l* 67.2%

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

    6. *-inverses 67.2%

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

    7. *-rgt-identity 67.2%

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

    8. exp-1-e 67.2%

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

    9. associate-/l* 67.2%

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

    10. associate-/l* 67.2%

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

    11. *-inverses 67.2%

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

    12. *-rgt-identity 67.2%

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

    13. *-commutative 67.2%

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

15. Simplified 67.2%

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

16. Final simplification 67.2%

    \[\leadsto \ell - \ell \cdot w \]
17. Add Preprocessing

Alternative 7: 56.9% 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 60.9%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))