exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 18.9s

Alternatives: 10

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. Final simplification 99.8%

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

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. associate-*l/ 99.8%

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

   3. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 97.7% 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. associate-*l/ 99.8%

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

   3. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in w around 0 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 82.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.021:\\ \;\;\;\;\ell + \left(\ell \cdot \left(w \cdot w\right) - w \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\ell}}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.021) (+ l (- (* l (* w w)) (* w l))) (/ (/ (* l l) l) (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.021) {
		tmp = l + ((l * (w * w)) - (w * l));
	} else {
		tmp = ((l * l) / 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 <= 0.021d0) then
        tmp = l + ((l * (w * w)) - (w * l))
    else
        tmp = ((l * l) / l) / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.021) {
		tmp = l + ((l * (w * w)) - (w * l));
	} else {
		tmp = ((l * l) / l) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.021:
		tmp = l + ((l * (w * w)) - (w * l))
	else:
		tmp = ((l * l) / l) / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.021)
		tmp = Float64(l + Float64(Float64(l * Float64(w * w)) - Float64(w * l)));
	else
		tmp = Float64(Float64(Float64(l * l) / l) / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.021)
		tmp = l + ((l * (w * w)) - (w * l));
	else
		tmp = ((l * l) / l) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.021], N[(l + N[(N[(l * N[(w * w), $MachinePrecision]), $MachinePrecision] - N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / l), $MachinePrecision] / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.0210000000000000013

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. exp-neg 99.8%

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

      3. div-inv 99.8%

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

      4. add-cube-cbrt 99.8%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 99.8%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 99.8%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 99.8%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 66.6%

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

   7. Taylor expanded in w around 0 66.2%

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

   8. Taylor expanded in w around 0 88.2%

      \[\leadsto \color{blue}{\ell + \left(-1 \cdot \left(\ell \cdot w\right) + \ell \cdot {w}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 88.2%

         \[\leadsto \ell + \color{blue}{\left(\ell \cdot {w}^{2} + -1 \cdot \left(\ell \cdot w\right)\right)} \]

      2. mul-1-neg 88.2%

         \[\leadsto \ell + \left(\ell \cdot {w}^{2} + \color{blue}{\left(-\ell \cdot w\right)}\right) \]

      3. unsub-neg 88.2%

         \[\leadsto \ell + \color{blue}{\left(\ell \cdot {w}^{2} - \ell \cdot w\right)} \]

      4. unpow2 88.2%

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

   10. Simplified 88.2%

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

   if 0.0210000000000000013 < w

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-cube-cbrt 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 100.0%

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

   7. Taylor expanded in w around 0 5.2%

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

   9. Applied egg-rr 53.8%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.021:\\ \;\;\;\;\ell + \left(\ell \cdot \left(w \cdot w\right) - w \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\ell}}{w + 1}\\ \end{array} \]

Alternative 5: 84.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -29:\\ \;\;\;\;\ell + \left(\ell \cdot \left(w \cdot w\right) - w \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(w + 1\right) + \left(w \cdot w\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -29.0)
   (+ l (- (* l (* w w)) (* w l)))
   (/ l (+ (+ w 1.0) (* (* w w) 0.5)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -29.0) {
		tmp = l + ((l * (w * w)) - (w * l));
	} else {
		tmp = l / ((w + 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 <= (-29.0d0)) then
        tmp = l + ((l * (w * w)) - (w * l))
    else
        tmp = l / ((w + 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 <= -29.0) {
		tmp = l + ((l * (w * w)) - (w * l));
	} else {
		tmp = l / ((w + 1.0) + ((w * w) * 0.5));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -29.0:
		tmp = l + ((l * (w * w)) - (w * l))
	else:
		tmp = l / ((w + 1.0) + ((w * w) * 0.5))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -29.0)
		tmp = Float64(l + Float64(Float64(l * Float64(w * w)) - Float64(w * l)));
	else
		tmp = Float64(l / Float64(Float64(w + 1.0) + Float64(Float64(w * w) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -29.0)
		tmp = l + ((l * (w * w)) - (w * l));
	else
		tmp = l / ((w + 1.0) + ((w * w) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -29.0], N[(l + N[(N[(l * N[(w * w), $MachinePrecision]), $MachinePrecision] - N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[(w + 1.0), $MachinePrecision] + N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -29

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-cube-cbrt 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 1.1%

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

   7. Taylor expanded in w around 0 1.0%

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

   8. Taylor expanded in w around 0 68.1%

      \[\leadsto \color{blue}{\ell + \left(-1 \cdot \left(\ell \cdot w\right) + \ell \cdot {w}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 68.1%

         \[\leadsto \ell + \color{blue}{\left(\ell \cdot {w}^{2} + -1 \cdot \left(\ell \cdot w\right)\right)} \]

      2. mul-1-neg 68.1%

         \[\leadsto \ell + \left(\ell \cdot {w}^{2} + \color{blue}{\left(-\ell \cdot w\right)}\right) \]

      3. unsub-neg 68.1%

         \[\leadsto \ell + \color{blue}{\left(\ell \cdot {w}^{2} - \ell \cdot w\right)} \]

      4. unpow2 68.1%

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

   10. Simplified 68.1%

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

   if -29 < w

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. exp-neg 99.7%

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

      3. div-inv 99.7%

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

      4. add-cube-cbrt 99.7%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 99.7%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 99.7%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 99.0%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + \left(w + 0.5 \cdot {w}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-+r+ 99.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(1 + w\right) + 0.5 \cdot {w}^{2}}} \]

      2. *-commutative 99.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\left(1 + w\right) + \color{blue}{{w}^{2} \cdot 0.5}} \]

      3. unpow2 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in w around 0 92.1%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -29:\\ \;\;\;\;\ell + \left(\ell \cdot \left(w \cdot w\right) - w \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(w + 1\right) + \left(w \cdot w\right) \cdot 0.5}\\ \end{array} \]

Alternative 6: 71.8% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.024:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\ell}}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.024) (- l (* w l)) (/ (/ (* l l) l) (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.024) {
		tmp = l - (w * l);
	} else {
		tmp = ((l * l) / 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 <= 0.024d0) then
        tmp = l - (w * l)
    else
        tmp = ((l * l) / l) / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.024) {
		tmp = l - (w * l);
	} else {
		tmp = ((l * l) / l) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.024:
		tmp = l - (w * l)
	else:
		tmp = ((l * l) / l) / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.024)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(Float64(Float64(l * l) / l) / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.024)
		tmp = l - (w * l);
	else
		tmp = ((l * l) / l) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.024], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / l), $MachinePrecision] / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.024

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. exp-neg 99.8%

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

      3. div-inv 99.8%

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

      4. add-cube-cbrt 99.8%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 99.8%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 99.8%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 99.8%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 66.6%

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

   7. Taylor expanded in w around 0 66.2%

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

   8. Taylor expanded in w around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   10. Simplified 73.8%

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

   if 0.024 < w

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-cube-cbrt 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 100.0%

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

   7. Taylor expanded in w around 0 5.2%

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

   9. Applied egg-rr 53.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.024:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\ell}}{w + 1}\\ \end{array} \]

Alternative 7: 69.7% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.112:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.112) (- l (* w l)) (/ l w)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.112) {
		tmp = l - (w * l);
	} else {
		tmp = l / w;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.112d0) then
        tmp = l - (w * l)
    else
        tmp = l / w
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.112) {
		tmp = l - (w * l);
	} else {
		tmp = l / w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.112:
		tmp = l - (w * l)
	else:
		tmp = l / w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.112)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(l / w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.112)
		tmp = l - (w * l);
	else
		tmp = l / w;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.112], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(l / w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.112000000000000002

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. exp-neg 99.8%

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

      3. div-inv 99.8%

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

      4. add-cube-cbrt 99.8%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 99.8%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 99.8%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 99.8%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 66.6%

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

   7. Taylor expanded in w around 0 66.2%

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

   8. Taylor expanded in w around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   10. Simplified 73.8%

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

   if 0.112000000000000002 < w

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-cube-cbrt 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 100.0%

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

   7. Taylor expanded in w around 0 40.4%

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

   8. Taylor expanded in w around inf 40.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.112:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \]

Alternative 8: 69.7% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.42:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.42) (- l (* w l)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.42) {
		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 <= (-0.42d0)) 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 <= -0.42) {
		tmp = l - (w * l);
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.42:
		tmp = l - (w * l)
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.42)
		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 <= -0.42)
		tmp = l - (w * l);
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.42], 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 < -0.419999999999999984

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-cube-cbrt 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 1.1%

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

   7. Taylor expanded in w around 0 1.0%

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

   8. Taylor expanded in w around 0 23.9%

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

      1. mul-1-neg 23.9%

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

      2. unsub-neg 23.9%

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

   10. Simplified 23.9%

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

   if -0.419999999999999984 < w

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. exp-neg 99.7%

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

      3. div-inv 99.7%

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

      4. add-cube-cbrt 99.7%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 99.7%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 99.7%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 99.7%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 99.3%

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

   7. Taylor expanded in w around 0 88.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.42:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]

Alternative 9: 63.1% accurate, 60.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 115000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 115000.0) l (/ l w)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 115000.0) {
		tmp = l;
	} else {
		tmp = l / w;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 115000.0d0) then
        tmp = l
    else
        tmp = l / w
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 115000.0) {
		tmp = l;
	} else {
		tmp = l / w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 115000.0:
		tmp = l
	else:
		tmp = l / w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 115000.0)
		tmp = l;
	else
		tmp = Float64(l / w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 115000.0)
		tmp = l;
	else
		tmp = l / w;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 115000.0], l, N[(l / w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 115000

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. exp-neg 99.8%

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

      3. div-inv 99.8%

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

      4. add-cube-cbrt 98.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\ell}^{\left(e^{w}\right)}} \cdot \sqrt[3]{{\ell}^{\left(e^{w}\right)}}\right) \cdot \sqrt[3]{{\ell}^{\left(e^{w}\right)}}}}{e^{w}} \]

      5. associate-/l* 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\ell}^{\left(e^{w}\right)}} \cdot \sqrt[3]{{\ell}^{\left(e^{w}\right)}}}{\frac{e^{w}}{\sqrt[3]{{\ell}^{\left(e^{w}\right)}}}}} \]

      6. pow2 98.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\ell}^{\left(e^{w}\right)}}\right)}^{2}}}{\frac{e^{w}}{\sqrt[3]{{\ell}^{\left(e^{w}\right)}}}} \]

   3. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{\ell}^{\left(e^{w}\right)}}\right)}^{2}}{\frac{e^{w}}{\sqrt[3]{{\ell}^{\left(e^{w}\right)}}}}} \]

   4. Taylor expanded in w around 0 67.1%

      \[\leadsto \color{blue}{\ell} \]

   if 115000 < w

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-cube-cbrt 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      5. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

      6. cbrt-unprod 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{e^{w} \cdot e^{w}}}}}{\sqrt[3]{e^{w}}} \]

      7. prod-exp 100.0%

         \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{e^{w + w}}}}}{\sqrt[3]{e^{w}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w + w}}}}{\sqrt[3]{e^{w}}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w + w}}}} \]

   6. Taylor expanded in w around 0 100.0%

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

   7. Taylor expanded in w around 0 40.4%

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

   8. Taylor expanded in w around inf 40.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 115000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w}\\ \end{array} \]

Alternative 10: 56.7% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. exp-neg 99.8%

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

   3. div-inv 99.8%

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

   4. add-cube-cbrt 98.5%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\ell}^{\left(e^{w}\right)}} \cdot \sqrt[3]{{\ell}^{\left(e^{w}\right)}}\right) \cdot \sqrt[3]{{\ell}^{\left(e^{w}\right)}}}}{e^{w}} \]

   5. associate-/l* 98.6%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\ell}^{\left(e^{w}\right)}} \cdot \sqrt[3]{{\ell}^{\left(e^{w}\right)}}}{\frac{e^{w}}{\sqrt[3]{{\ell}^{\left(e^{w}\right)}}}}} \]

   6. pow2 98.6%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\ell}^{\left(e^{w}\right)}}\right)}^{2}}}{\frac{e^{w}}{\sqrt[3]{{\ell}^{\left(e^{w}\right)}}}} \]

3. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{\ell}^{\left(e^{w}\right)}}\right)}^{2}}{\frac{e^{w}}{\sqrt[3]{{\ell}^{\left(e^{w}\right)}}}}} \]

4. Taylor expanded in w around 0 59.9%

   \[\leadsto \color{blue}{\ell} \]

5. Final simplification 59.9%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023278 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))