exp-w (used to crash)

Percentage Accurate: 99.5% → 99.3%

Time: 12.1s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -222:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{w \cdot \left(w \cdot 0.5 + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -222.0)
   (/ l (exp w))
   (/ (pow l (exp w)) (+ (* w (+ (* w 0.5) 1.0)) 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -222.0) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, exp(w)) / ((w * ((w * 0.5) + 1.0)) + 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 <= (-222.0d0)) then
        tmp = l / exp(w)
    else
        tmp = (l ** exp(w)) / ((w * ((w * 0.5d0) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -222.0) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / ((w * ((w * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -222.0:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) / ((w * ((w * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -222.0)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ exp(w)) / Float64(Float64(w * Float64(Float64(w * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -222.0)
		tmp = l / exp(w);
	else
		tmp = (l ^ exp(w)) / ((w * ((w * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -222.0], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[(w * N[(N[(w * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -222

   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%

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

   6. Taylor expanded in l around 0 100.0%

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

   if -222 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -222:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{w \cdot \left(w \cdot 0.5 + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. exp-neg 99.4%

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

   2. remove-double-neg 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. remove-double-neg 99.4%

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

3. Simplified 99.4%

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

Alternative 4: 99.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in l around 0 100.0%

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

   if -1 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0 98.8%

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

      1. *-commutative 99.2%

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

   10. Simplified 98.8%

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

4. Final simplification 99.1%

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

Alternative 5: 97.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. exp-neg 99.4%

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

   2. remove-double-neg 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. remove-double-neg 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in w around 0 97.4%

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

6. Taylor expanded in l around 0 97.4%

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

Alternative 6: 86.1% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   1. Initial program 100.0%

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

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

   6. Taylor expanded in w around 0 72.6%

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

   7. Taylor expanded in l around 0 72.6%

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

   if -1.6000000000000001 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 96.5%

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

   6. Taylor expanded in l around 0 96.5%

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

   7. Taylor expanded in w around 0 93.3%

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

      1. *-commutative 93.3%

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

   9. Simplified 93.3%

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

4. Final simplification 88.0%

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

Alternative 7: 82.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 350000000:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w \cdot \left(w \cdot \left(0.5 + w \cdot 0.16666666666666666\right) + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 350000000.0)
   (- l (* w (+ l (* w (* l -0.5)))))
   (/ l (+ (* w (+ (* w (+ 0.5 (* w 0.16666666666666666))) 1.0)) 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 350000000.0) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = l / ((w * ((w * (0.5 + (w * 0.16666666666666666))) + 1.0)) + 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 <= 350000000.0d0) then
        tmp = l - (w * (l + (w * (l * (-0.5d0)))))
    else
        tmp = l / ((w * ((w * (0.5d0 + (w * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 350000000.0) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = l / ((w * ((w * (0.5 + (w * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 350000000.0:
		tmp = l - (w * (l + (w * (l * -0.5))))
	else:
		tmp = l / ((w * ((w * (0.5 + (w * 0.16666666666666666))) + 1.0)) + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 350000000.0)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(l * -0.5)))));
	else
		tmp = Float64(l / Float64(Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 350000000.0)
		tmp = l - (w * (l + (w * (l * -0.5))));
	else
		tmp = l / ((w * ((w * (0.5 + (w * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 350000000.0], N[(l - N[(w * N[(l + N[(w * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[(w * N[(N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.5e8

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 96.8%

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

   6. Taylor expanded in w around 0 86.0%

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

      1. associate-*r* 86.0%

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

      2. neg-mul-1 86.0%

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

      3. distribute-rgt-out 86.0%

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

      4. metadata-eval 86.0%

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

   8. Simplified 86.0%

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

   if 3.5e8 < 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%

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

   6. Taylor expanded in l around 0 100.0%

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

   7. Taylor expanded in w around 0 86.6%

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

      1. *-commutative 86.6%

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

   9. Simplified 86.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 350000000:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w \cdot \left(w \cdot \left(0.5 + w \cdot 0.16666666666666666\right) + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 520000000:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w \cdot \left(w \cdot 0.5 + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 520000000.0)
   (- l (* w (+ l (* w (* l -0.5)))))
   (/ l (+ (* w (+ (* w 0.5) 1.0)) 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 520000000.0) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 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 <= 520000000.0d0) then
        tmp = l - (w * (l + (w * (l * (-0.5d0)))))
    else
        tmp = l / ((w * ((w * 0.5d0) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 520000000.0) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 520000000.0:
		tmp = l - (w * (l + (w * (l * -0.5))))
	else:
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 520000000.0)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(l * -0.5)))));
	else
		tmp = Float64(l / Float64(Float64(w * Float64(Float64(w * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 520000000.0)
		tmp = l - (w * (l + (w * (l * -0.5))));
	else
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 520000000.0], N[(l - N[(w * N[(l + N[(w * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[(w * N[(N[(w * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 5.2e8

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 96.8%

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

   6. Taylor expanded in w around 0 86.0%

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

      1. associate-*r* 86.0%

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

      2. neg-mul-1 86.0%

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

      3. distribute-rgt-out 86.0%

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

      4. metadata-eval 86.0%

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

   8. Simplified 86.0%

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

   if 5.2e8 < 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%

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

   6. Taylor expanded in l around 0 100.0%

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

   7. Taylor expanded in w around 0 72.2%

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

      1. *-commutative 100.0%

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

   9. Simplified 72.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 520000000:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w \cdot \left(w \cdot 0.5 + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -650:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w \cdot \left(w \cdot 0.5 + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -650.0) (- l (* w l)) (/ l (+ (* w (+ (* w 0.5) 1.0)) 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -650.0) {
		tmp = l - (w * l);
	} else {
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 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 <= (-650.0d0)) then
        tmp = l - (w * l)
    else
        tmp = l / ((w * ((w * 0.5d0) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -650.0) {
		tmp = l - (w * l);
	} else {
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -650.0:
		tmp = l - (w * l)
	else:
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -650.0)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(l / Float64(Float64(w * Float64(Float64(w * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -650.0)
		tmp = l - (w * l);
	else
		tmp = l / ((w * ((w * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -650.0], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[(w * N[(N[(w * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -650

   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%

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

   6. Taylor expanded in l around 0 100.0%

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

   7. Taylor expanded in w around 0 34.3%

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

      1. mul-1-neg 34.3%

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

      2. unsub-neg 34.3%

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

   9. Simplified 34.3%

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

   if -650 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 96.5%

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

   6. Taylor expanded in l around 0 96.5%

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

   7. Taylor expanded in w around 0 89.8%

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

      1. *-commutative 99.2%

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

   9. Simplified 89.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -650:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w \cdot \left(w \cdot 0.5 + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.62:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.62) (- l (* w l)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.62) {
		tmp = l - (w * l);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.62d0)) then
        tmp = l - (w * l)
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.62) {
		tmp = l - (w * l);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.62:
		tmp = l - (w * l)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.62)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.62)
		tmp = l - (w * l);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.62], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.619999999999999996

   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%

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

   6. Taylor expanded in l around 0 100.0%

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

   7. Taylor expanded in w around 0 34.3%

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

      1. mul-1-neg 34.3%

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

      2. unsub-neg 34.3%

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

   9. Simplified 34.3%

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

   if -0.619999999999999996 < w

   1. Initial program 99.1%

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

      1. exp-neg 99.1%

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

      2. remove-double-neg 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-lft-identity 99.1%

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

      5. remove-double-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0 96.5%

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

   6. Taylor expanded in l around 0 96.5%

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

   7. Taylor expanded in w around 0 84.6%

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

      1. +-commutative 98.8%

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

   9. Simplified 84.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.62:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.4% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. exp-neg 99.4%

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

   2. remove-double-neg 99.4%

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

   3. associate-*l/ 99.4%

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

   4. *-lft-identity 99.4%

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

   5. remove-double-neg 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in w around 0 97.4%

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

6. Taylor expanded in l around 0 97.4%

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

7. Taylor expanded in w around 0 63.4%

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

   1. mul-1-neg 63.4%

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

   2. unsub-neg 63.4%

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

9. Simplified 63.4%

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

10. Final simplification 63.4%

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

Alternative 12: 57.1% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in w around 0 56.1%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))