exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 38.8s

Alternatives: 13

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: 99.5% 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 99.8%

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

Alternative 2: 99.5% 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. Taylor expanded in w around 0 98.3%

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

Alternative 4: 86.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.011:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\left(\ell - \ell \cdot 0.5\right) + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 - w \cdot \left(-1 - w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.011)
   (+
    l
    (*
     w
     (-
      (*
       w
       (+
        (- l (* l 0.5))
        (* w (- (- (* l 0.5) l) (+ (* l -0.5) (* l 0.16666666666666666))))))
      l)))
   (/ l (- 1.0 (* w (- -1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.011) {
		tmp = l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l));
	} else {
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.011d0)) then
        tmp = l + (w * ((w * ((l - (l * 0.5d0)) + (w * (((l * 0.5d0) - l) - ((l * (-0.5d0)) + (l * 0.16666666666666666d0)))))) - l))
    else
        tmp = l / (1.0d0 - (w * ((-1.0d0) - (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.011) {
		tmp = l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l));
	} else {
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.011:
		tmp = l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l))
	else:
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.011)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(Float64(l - Float64(l * 0.5)) + Float64(w * Float64(Float64(Float64(l * 0.5) - l) - Float64(Float64(l * -0.5) + Float64(l * 0.16666666666666666)))))) - l)));
	else
		tmp = Float64(l / Float64(1.0 - Float64(w * Float64(-1.0 - Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.011)
		tmp = l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l));
	else
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.011], N[(l + N[(w * N[(N[(w * N[(N[(l - N[(l * 0.5), $MachinePrecision]), $MachinePrecision] + N[(w * N[(N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision] - N[(N[(l * -0.5), $MachinePrecision] + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 - N[(w * N[(-1.0 - N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.010999999999999999

   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 98.7%

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

   6. Taylor expanded in w around 0 64.4%

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

   if -0.010999999999999999 < 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 98.1%

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

   6. Taylor expanded in w around 0 93.0%

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

      1. *-commutative 93.0%

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

   8. Simplified 93.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.011:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\left(\ell - \ell \cdot 0.5\right) + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 - w \cdot \left(-1 - w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 - w \cdot \left(-1 - w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.17)
   (+ l (* l (* w (+ -1.0 (* w 0.5)))))
   (/ l (- 1.0 (* w (- -1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.17d0) then
        tmp = l + (l * (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = l / (1.0d0 - (w * ((-1.0d0) - (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.17:
		tmp = l + (l * (w * (-1.0 + (w * 0.5))))
	else:
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.17)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(l / Float64(1.0 - Float64(w * Float64(-1.0 - Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.17)
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	else
		tmp = l / (1.0 - (w * (-1.0 - (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.17], N[(l + N[(l * 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 * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.170000000000000012

   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 97.9%

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

   6. Taylor expanded in w around 0 80.5%

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

      1. associate-*r* 80.5%

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

      2. neg-mul-1 80.5%

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

      3. distribute-rgt-out 80.5%

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

      4. metadata-eval 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in l around 0 83.5%

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

   if 0.170000000000000012 < 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 78.7%

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

      1. *-commutative 78.7%

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

   8. Simplified 78.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 - w \cdot \left(-1 - w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.25 \cdot 10^{-12}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.25e-12)
   (+ l (* l (* w (+ -1.0 (* w 0.5)))))
   (/ l (+ 1.0 (* w (* w 0.5))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.25e-12) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = l / (1.0 + (w * (w * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.25d-12)) then
        tmp = l + (l * (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = l / (1.0d0 + (w * (w * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.25e-12) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = l / (1.0 + (w * (w * 0.5)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.25e-12:
		tmp = l + (l * (w * (-1.0 + (w * 0.5))))
	else:
		tmp = l / (1.0 + (w * (w * 0.5)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.25e-12)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(w * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.25e-12)
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	else
		tmp = l / (1.0 + (w * (w * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.25e-12], N[(l + N[(l * N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.24999999999999992e-12

   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 97.2%

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

   6. Taylor expanded in w around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. neg-mul-1 47.2%

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

      3. distribute-rgt-out 47.2%

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

      4. metadata-eval 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in l around 0 56.0%

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

   if -1.24999999999999992e-12 < w

   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 98.7%

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

   6. Taylor expanded in w around 0 92.8%

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

      1. *-commutative 92.8%

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

   8. Simplified 92.8%

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

   9. Taylor expanded in w around inf 92.8%

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

       1. *-commutative 92.8%

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

   11. Simplified 92.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.25 \cdot 10^{-12}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.086:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.086) (+ l (* w (* w (* l 0.5)))) (/ l (+ 1.0 (* w (* w 0.5))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.086) {
		tmp = l + (w * (w * (l * 0.5)));
	} else {
		tmp = l / (1.0 + (w * (w * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.086d0) then
        tmp = l + (w * (w * (l * 0.5d0)))
    else
        tmp = l / (1.0d0 + (w * (w * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.086) {
		tmp = l + (w * (w * (l * 0.5)));
	} else {
		tmp = l / (1.0 + (w * (w * 0.5)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.086:
		tmp = l + (w * (w * (l * 0.5)))
	else:
		tmp = l / (1.0 + (w * (w * 0.5)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.086)
		tmp = Float64(l + Float64(w * Float64(w * Float64(l * 0.5))));
	else
		tmp = Float64(l / Float64(1.0 + Float64(w * Float64(w * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.086)
		tmp = l + (w * (w * (l * 0.5)));
	else
		tmp = l / (1.0 + (w * (w * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.086], N[(l + N[(w * N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(1.0 + N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.085999999999999993

   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 97.9%

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

   6. Taylor expanded in w around 0 80.5%

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

      1. associate-*r* 80.5%

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

      2. neg-mul-1 80.5%

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

      3. distribute-rgt-out 80.5%

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

      4. metadata-eval 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in w around inf 80.5%

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

       1. associate-*r* 80.5%

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

       2. *-commutative 80.5%

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

   11. Simplified 80.5%

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

   if 0.085999999999999993 < 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 75.5%

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

      1. *-commutative 75.5%

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

   8. Simplified 75.5%

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

   9. Taylor expanded in w around inf 75.5%

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

       1. *-commutative 75.5%

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

   11. Simplified 75.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.086:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.04d0) then
        tmp = l + (w * (w * (l * 0.5d0)))
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, 0.04], N[(l + N[(w * N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.0400000000000000008

   1. Initial program 99.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. 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 97.9%

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

   6. Taylor expanded in w around 0 80.5%

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

      1. associate-*r* 80.5%

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

      2. neg-mul-1 80.5%

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

      3. distribute-rgt-out 80.5%

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

      4. metadata-eval 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in w around inf 80.5%

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

       1. associate-*r* 80.5%

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

       2. *-commutative 80.5%

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

   11. Simplified 80.5%

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

   if 0.0400000000000000008 < 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 55.2%

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

      1. +-commutative 55.2%

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

   8. Simplified 55.2%

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

4. Final simplification 76.1%

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

Alternative 9: 70.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.00078:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.00078) (* l (- 1.0 w)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.00078) {
		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 <= (-0.00078d0)) 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 <= -0.00078) {
		tmp = l * (1.0 - w);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.00078:
		tmp = l * (1.0 - w)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.00078)
		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 <= -0.00078)
		tmp = l * (1.0 - w);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.00078], 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 < -7.79999999999999986e-4

   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 98.7%

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

   6. Taylor expanded in w around 0 23.0%

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

      1. mul-1-neg 23.0%

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

      2. unsub-neg 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in l around 0 23.0%

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

   if -7.79999999999999986e-4 < 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 98.1%

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

   6. Taylor expanded in w around 0 87.4%

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

      1. +-commutative 87.4%

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

   8. Simplified 87.4%

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

Alternative 10: 64.0% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.012:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.012) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.012) {
		tmp = w * -l;
	} 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 <= (-0.012d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.012) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.012:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.012)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.012)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.012], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.012

   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 98.7%

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

   6. Taylor expanded in w around 0 23.0%

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

      1. mul-1-neg 23.0%

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

      2. unsub-neg 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in w around inf 23.0%

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

       1. associate-*r* 23.0%

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

       2. neg-mul-1 23.0%

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

       3. *-commutative 23.0%

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

   11. Simplified 23.0%

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

   if -0.012 < 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 98.1%

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

   6. Taylor expanded in w around 0 75.6%

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

Alternative 11: 63.7% 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 99.8%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 60.4%

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

   1. mul-1-neg 60.4%

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

   2. unsub-neg 60.4%

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

8. Simplified 60.4%

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

9. Final simplification 60.4%

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

Alternative 12: 63.7% 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. 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 60.4%

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

   1. mul-1-neg 60.4%

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

   2. unsub-neg 60.4%

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

8. Simplified 60.4%

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

9. Taylor expanded in l around 0 60.4%

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

Alternative 13: 56.4% 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 55.4%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))