exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 17.4s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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 99.3%

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

3. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]
6. 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 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 96.8%

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

6. Final simplification 96.8%

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

Alternative 4: 86.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.75:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(w \cdot \left(\ell \cdot 1.1666666666666667\right) + \left(\ell \cdot 0.5 - \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} + w \cdot \left(0.5 \cdot \frac{-1}{\ell} - 0.16666666666666666 \cdot \frac{w}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.75)
   (- l (* w (+ l (* w (+ (* w (* l 1.1666666666666667)) (- (* l 0.5) l))))))
   (/
    -1.0
    (+
     (/ -1.0 l)
     (*
      w
      (+
       (/ -1.0 l)
       (* w (- (* 0.5 (/ -1.0 l)) (* 0.16666666666666666 (/ w l))))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.75) {
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) + (w * ((0.5 * (-1.0 / l)) - (0.16666666666666666 * (w / 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 <= 0.75d0) then
        tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667d0)) + ((l * 0.5d0) - l)))))
    else
        tmp = (-1.0d0) / (((-1.0d0) / l) + (w * (((-1.0d0) / l) + (w * ((0.5d0 * ((-1.0d0) / l)) - (0.16666666666666666d0 * (w / l)))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.75) {
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) + (w * ((0.5 * (-1.0 / l)) - (0.16666666666666666 * (w / l)))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.75:
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))))
	else:
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) + (w * ((0.5 * (-1.0 / l)) - (0.16666666666666666 * (w / l)))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.75)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(Float64(w * Float64(l * 1.1666666666666667)) + Float64(Float64(l * 0.5) - l))))));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / l) + Float64(w * Float64(Float64(-1.0 / l) + Float64(w * Float64(Float64(0.5 * Float64(-1.0 / l)) - Float64(0.16666666666666666 * Float64(w / l))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.75)
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))));
	else
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) + (w * ((0.5 * (-1.0 / l)) - (0.16666666666666666 * (w / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.75], N[(l - N[(w * N[(l + N[(w * N[(N[(w * N[(l * 1.1666666666666667), $MachinePrecision]), $MachinePrecision] + N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / l), $MachinePrecision] + N[(w * N[(N[(-1.0 / l), $MachinePrecision] + N[(w * N[(N[(0.5 * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * N[(w / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.75

   1. Initial program 99.6%

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

      1. exp-neg 99.6%

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

      2. remove-double-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. remove-double-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 88.1%

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

      1. associate-+r+ 88.1%

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

      2. distribute-lft-in 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. distribute-lft-out 88.1%

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

      2. distribute-lft-out 88.1%

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

      3. metadata-eval 88.1%

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

      4. distribute-lft-out 88.1%

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

      5. metadata-eval 88.1%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{1.1666666666666667}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   10. Simplified 88.1%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \color{blue}{\left(w \cdot \left(\ell \cdot 1.1666666666666667\right)\right)} - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   if 0.75 < w

   1. Initial program 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. metadata-eval 96.8%

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

      3. metadata-eval 96.8%

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

      4. metadata-eval 96.8%

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

      5. pow-prod-up 9.7%

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

      6. pow1 9.7%

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

      7. add-sqr-sqrt 9.7%

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

      8. sqrt-unprod 9.7%

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

      9. sqr-neg 9.7%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 96.8%

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

      12. pow1 96.8%

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

      13. inv-pow 96.8%

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

      14. exp-neg 96.8%

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

      15. pow1 96.8%

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

      16. sqrt-unprod 96.8%

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

      17. add-sqr-sqrt 96.8%

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

      18. exp-neg 96.8%

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

      19. associate-/r/ 96.8%

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

   4. Applied egg-rr 87.7%

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

      1. div-inv 87.7%

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

      2. exp-neg 87.7%

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

      3. associate-/l/ 87.7%

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

   6. Applied egg-rr 87.7%

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

   7. Taylor expanded in w around 0 72.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.75:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(w \cdot \left(\ell \cdot 1.1666666666666667\right) + \left(\ell \cdot 0.5 - \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} + w \cdot \left(0.5 \cdot \frac{-1}{\ell} - 0.16666666666666666 \cdot \frac{w}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.68:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(w \cdot \left(\ell \cdot 1.1666666666666667\right) + \left(\ell \cdot 0.5 - \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} - 0.5 \cdot \frac{w}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.68)
   (- l (* w (+ l (* w (+ (* w (* l 1.1666666666666667)) (- (* l 0.5) l))))))
   (/ -1.0 (+ (/ -1.0 l) (* w (- (/ -1.0 l) (* 0.5 (/ w l))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.68) {
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / 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 <= 0.68d0) then
        tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667d0)) + ((l * 0.5d0) - l)))))
    else
        tmp = (-1.0d0) / (((-1.0d0) / l) + (w * (((-1.0d0) / l) - (0.5d0 * (w / l)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.68) {
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.68:
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))))
	else:
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.68)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(Float64(w * Float64(l * 1.1666666666666667)) + Float64(Float64(l * 0.5) - l))))));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / l) + Float64(w * Float64(Float64(-1.0 / l) - Float64(0.5 * Float64(w / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.68)
		tmp = l - (w * (l + (w * ((w * (l * 1.1666666666666667)) + ((l * 0.5) - l)))));
	else
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.68], N[(l - N[(w * N[(l + N[(w * N[(N[(w * N[(l * 1.1666666666666667), $MachinePrecision]), $MachinePrecision] + N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / l), $MachinePrecision] + N[(w * N[(N[(-1.0 / l), $MachinePrecision] - N[(0.5 * N[(w / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.680000000000000049

   1. Initial program 99.6%

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

      1. exp-neg 99.6%

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

      2. remove-double-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. remove-double-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 88.1%

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

      1. associate-+r+ 88.1%

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

      2. distribute-lft-in 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. distribute-lft-out 88.1%

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

      2. distribute-lft-out 88.1%

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

      3. metadata-eval 88.1%

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

      4. distribute-lft-out 88.1%

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

      5. metadata-eval 88.1%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{1.1666666666666667}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   10. Simplified 88.1%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \color{blue}{\left(w \cdot \left(\ell \cdot 1.1666666666666667\right)\right)} - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   if 0.680000000000000049 < w

   1. Initial program 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. metadata-eval 96.8%

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

      3. metadata-eval 96.8%

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

      4. metadata-eval 96.8%

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

      5. pow-prod-up 9.7%

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

      6. pow1 9.7%

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

      7. add-sqr-sqrt 9.7%

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

      8. sqrt-unprod 9.7%

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

      9. sqr-neg 9.7%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 96.8%

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

      12. pow1 96.8%

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

      13. inv-pow 96.8%

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

      14. exp-neg 96.8%

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

      15. pow1 96.8%

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

      16. sqrt-unprod 96.8%

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

      17. add-sqr-sqrt 96.8%

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

      18. exp-neg 96.8%

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

      19. associate-/r/ 96.8%

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

   4. Applied egg-rr 87.7%

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

      1. div-inv 87.7%

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

      2. exp-neg 87.7%

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

      3. associate-/l/ 87.7%

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

   6. Applied egg-rr 87.7%

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

   7. Taylor expanded in w around 0 66.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.68:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(w \cdot \left(\ell \cdot 1.1666666666666667\right) + \left(\ell \cdot 0.5 - \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} - 0.5 \cdot \frac{w}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 250000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} - 0.5 \cdot \frac{w}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 250000.0)
   (* l (+ 1.0 (* w (+ -1.0 (* w 0.5)))))
   (/ -1.0 (+ (/ -1.0 l) (* w (- (/ -1.0 l) (* 0.5 (/ w l))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 250000.0) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / 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 <= 250000.0d0) then
        tmp = l * (1.0d0 + (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = (-1.0d0) / (((-1.0d0) / l) + (w * (((-1.0d0) / l) - (0.5d0 * (w / l)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 250000.0) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 250000.0:
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))))
	else:
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 250000.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / l) + Float64(w * Float64(Float64(-1.0 / l) - Float64(0.5 * Float64(w / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 250000.0)
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	else
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 250000.0], N[(l * N[(1.0 + N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / l), $MachinePrecision] + N[(w * N[(N[(-1.0 / l), $MachinePrecision] - N[(0.5 * N[(w / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.5e5

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 96.5%

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

   6. Taylor expanded in w around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

      3. distribute-rgt-out 81.2%

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

      4. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in l around 0 85.7%

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

   if 2.5e5 < w

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. pow1 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. sqr-neg 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. pow1 100.0%

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

      13. inv-pow 100.0%

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

      14. exp-neg 100.0%

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

      15. pow1 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. exp-neg 100.0%

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

      19. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. exp-neg 100.0%

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

      3. associate-/l/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 75.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 250000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} - 0.5 \cdot \frac{w}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.65:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5 + -0.16666666666666666 \cdot \left(w \cdot \ell\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} - 0.5 \cdot \frac{w}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.65)
   (+ l (* w (- (* w (+ (* l 0.5) (* -0.16666666666666666 (* w l)))) l)))
   (/ -1.0 (+ (/ -1.0 l) (* w (- (/ -1.0 l) (* 0.5 (/ w l))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.65) {
		tmp = l + (w * ((w * ((l * 0.5) + (-0.16666666666666666 * (w * l)))) - l));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / 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 <= 0.65d0) then
        tmp = l + (w * ((w * ((l * 0.5d0) + ((-0.16666666666666666d0) * (w * l)))) - l))
    else
        tmp = (-1.0d0) / (((-1.0d0) / l) + (w * (((-1.0d0) / l) - (0.5d0 * (w / l)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.65) {
		tmp = l + (w * ((w * ((l * 0.5) + (-0.16666666666666666 * (w * l)))) - l));
	} else {
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.65:
		tmp = l + (w * ((w * ((l * 0.5) + (-0.16666666666666666 * (w * l)))) - l))
	else:
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.65)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(Float64(l * 0.5) + Float64(-0.16666666666666666 * Float64(w * l)))) - l)));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / l) + Float64(w * Float64(Float64(-1.0 / l) - Float64(0.5 * Float64(w / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.65)
		tmp = l + (w * ((w * ((l * 0.5) + (-0.16666666666666666 * (w * l)))) - l));
	else
		tmp = -1.0 / ((-1.0 / l) + (w * ((-1.0 / l) - (0.5 * (w / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.65], N[(l + N[(w * N[(N[(w * N[(N[(l * 0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / l), $MachinePrecision] + N[(w * N[(N[(-1.0 / l), $MachinePrecision] - N[(0.5 * N[(w / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.650000000000000022

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. metadata-eval 99.6%

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

      3. metadata-eval 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow-prod-up 70.7%

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

      6. pow1 70.7%

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

      7. add-sqr-sqrt 34.9%

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

      8. sqrt-unprod 99.4%

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

      9. sqr-neg 99.4%

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

      10. sqrt-unprod 64.5%

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

      11. add-sqr-sqrt 98.5%

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

      12. pow1 98.5%

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

      13. inv-pow 98.5%

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

      14. exp-neg 98.5%

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

      15. pow1 98.5%

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

      16. sqrt-unprod 98.5%

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

      17. add-sqr-sqrt 98.5%

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

      18. exp-neg 98.5%

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

      19. associate-/r/ 98.3%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in w around 0 88.1%

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

   if 0.650000000000000022 < w

   1. Initial program 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. metadata-eval 96.8%

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

      3. metadata-eval 96.8%

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

      4. metadata-eval 96.8%

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

      5. pow-prod-up 9.7%

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

      6. pow1 9.7%

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

      7. add-sqr-sqrt 9.7%

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

      8. sqrt-unprod 9.7%

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

      9. sqr-neg 9.7%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 96.8%

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

      12. pow1 96.8%

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

      13. inv-pow 96.8%

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

      14. exp-neg 96.8%

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

      15. pow1 96.8%

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

      16. sqrt-unprod 96.8%

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

      17. add-sqr-sqrt 96.8%

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

      18. exp-neg 96.8%

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

      19. associate-/r/ 96.8%

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

   4. Applied egg-rr 87.7%

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

      1. div-inv 87.7%

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

      2. exp-neg 87.7%

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

      3. associate-/l/ 87.7%

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

   6. Applied egg-rr 87.7%

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

   7. Taylor expanded in w around 0 66.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.65:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5 + -0.16666666666666666 \cdot \left(w \cdot \ell\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} + w \cdot \left(\frac{-1}{\ell} - 0.5 \cdot \frac{w}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 360000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} - \frac{w}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 360000.0)
   (* l (+ 1.0 (* w (+ -1.0 (* w 0.5)))))
   (/ -1.0 (- (/ -1.0 l) (/ w l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 360000.0) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = -1.0 / ((-1.0 / l) - (w / 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 <= 360000.0d0) then
        tmp = l * (1.0d0 + (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = (-1.0d0) / (((-1.0d0) / l) - (w / l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 360000.0) {
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = -1.0 / ((-1.0 / l) - (w / l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 360000.0:
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))))
	else:
		tmp = -1.0 / ((-1.0 / l) - (w / l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 360000.0)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / l) - Float64(w / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 360000.0)
		tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
	else
		tmp = -1.0 / ((-1.0 / l) - (w / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 360000.0], N[(l * N[(1.0 + N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / l), $MachinePrecision] - N[(w / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.6e5

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 96.5%

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

   6. Taylor expanded in w around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

      3. distribute-rgt-out 81.2%

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

      4. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in l around 0 85.7%

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

   if 3.6e5 < w

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. pow1 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. sqr-neg 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. pow1 100.0%

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

      13. inv-pow 100.0%

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

      14. exp-neg 100.0%

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

      15. pow1 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. exp-neg 100.0%

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

      19. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. exp-neg 100.0%

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

      3. associate-/l/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 51.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 360000:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} - \frac{w}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 20500000:\\ \;\;\;\;\ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} - \frac{w}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 20500000.0)
   (+ l (* w (* 0.5 (* w l))))
   (/ -1.0 (- (/ -1.0 l) (/ w l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 20500000.0) {
		tmp = l + (w * (0.5 * (w * l)));
	} else {
		tmp = -1.0 / ((-1.0 / l) - (w / 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 <= 20500000.0d0) then
        tmp = l + (w * (0.5d0 * (w * l)))
    else
        tmp = (-1.0d0) / (((-1.0d0) / l) - (w / l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 20500000.0) {
		tmp = l + (w * (0.5 * (w * l)));
	} else {
		tmp = -1.0 / ((-1.0 / l) - (w / l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 20500000.0:
		tmp = l + (w * (0.5 * (w * l)))
	else:
		tmp = -1.0 / ((-1.0 / l) - (w / l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 20500000.0)
		tmp = Float64(l + Float64(w * Float64(0.5 * Float64(w * l))));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / l) - Float64(w / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 20500000.0)
		tmp = l + (w * (0.5 * (w * l)));
	else
		tmp = -1.0 / ((-1.0 / l) - (w / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 20500000.0], N[(l + N[(w * N[(0.5 * N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / l), $MachinePrecision] - N[(w / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.05e7

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. remove-double-neg 99.2%

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

      3. associate-*l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. remove-double-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in w around 0 96.5%

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

   6. Taylor expanded in w around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

      3. distribute-rgt-out 81.2%

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

      4. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in w around inf 81.2%

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

   if 2.05e7 < w

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. pow1 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. sqr-neg 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. pow1 100.0%

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

      13. inv-pow 100.0%

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

      14. exp-neg 100.0%

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

      15. pow1 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. exp-neg 100.0%

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

      19. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. exp-neg 100.0%

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

      3. associate-/l/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 51.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 20500000:\\ \;\;\;\;\ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\ell} - \frac{w}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.4% accurate, 33.9× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* w (* 0.5 (* w l)))))

C

double code(double w, double l) {
	return l + (w * (0.5 * (w * l)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * (0.5d0 * (w * l)))
end function

Java

public static double code(double w, double l) {
	return l + (w * (0.5 * (w * l)));
}

Python

def code(w, l):
	return l + (w * (0.5 * (w * l)))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(0.5 * Float64(w * l))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * (0.5 * (w * l)));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 96.8%

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

6. Taylor expanded in w around 0 72.9%

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

   1. associate-*r* 72.9%

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

   2. neg-mul-1 72.9%

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

   3. distribute-rgt-out 72.9%

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

   4. metadata-eval 72.9%

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

8. Simplified 72.9%

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

9. Taylor expanded in w around inf 72.9%

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

10. Final simplification 72.9%

    \[\leadsto \ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right)\right) \]
11. Add Preprocessing

Alternative 11: 63.2% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 96.8%

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

6. Taylor expanded in w around 0 69.1%

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

   1. mul-1-neg 69.1%

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

   2. unsub-neg 69.1%

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

8. Simplified 69.1%

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

9. Final simplification 69.1%

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

Alternative 12: 56.2% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. remove-double-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. *-lft-identity 99.3%

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

   5. remove-double-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 96.8%

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

6. Taylor expanded in w around 0 62.5%

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

7. Final simplification 62.5%

   \[\leadsto \ell \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))