exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 24.1s

Alternatives: 9

Speedup: N/A×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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 99.5%

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

   1. exp-neg 99.5%

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

   2. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.5 \cdot 10^{+239}:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + 1\right)\right)\right|\\ \mathbf{elif}\;w \leq -1.55 \cdot 10^{+109} \lor \neg \left(w \leq 2.5\right):\\ \;\;\;\;\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \left|\ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.5e+239)
   (fabs (* l (* w (+ (log l) 1.0))))
   (if (or (<= w -1.55e+109) (not (<= w 2.5)))
     (/ (* l (* w (log l))) (exp w))
     (+ l (fabs (* l (* w (+ (log l) -1.0))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.5e+239) {
		tmp = fabs((l * (w * (log(l) + 1.0))));
	} else if ((w <= -1.55e+109) || !(w <= 2.5)) {
		tmp = (l * (w * log(l))) / exp(w);
	} else {
		tmp = l + fabs((l * (w * (log(l) + -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 <= (-2.5d+239)) then
        tmp = abs((l * (w * (log(l) + 1.0d0))))
    else if ((w <= (-1.55d+109)) .or. (.not. (w <= 2.5d0))) then
        tmp = (l * (w * log(l))) / exp(w)
    else
        tmp = l + abs((l * (w * (log(l) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -2.5e+239) {
		tmp = Math.abs((l * (w * (Math.log(l) + 1.0))));
	} else if ((w <= -1.55e+109) || !(w <= 2.5)) {
		tmp = (l * (w * Math.log(l))) / Math.exp(w);
	} else {
		tmp = l + Math.abs((l * (w * (Math.log(l) + -1.0))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.5e+239:
		tmp = math.fabs((l * (w * (math.log(l) + 1.0))))
	elif (w <= -1.55e+109) or not (w <= 2.5):
		tmp = (l * (w * math.log(l))) / math.exp(w)
	else:
		tmp = l + math.fabs((l * (w * (math.log(l) + -1.0))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.5e+239)
		tmp = abs(Float64(l * Float64(w * Float64(log(l) + 1.0))));
	elseif ((w <= -1.55e+109) || !(w <= 2.5))
		tmp = Float64(Float64(l * Float64(w * log(l))) / exp(w));
	else
		tmp = Float64(l + abs(Float64(l * Float64(w * Float64(log(l) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -2.5e+239)
		tmp = abs((l * (w * (log(l) + 1.0))));
	elseif ((w <= -1.55e+109) || ~((w <= 2.5)))
		tmp = (l * (w * log(l))) / exp(w);
	else
		tmp = l + abs((l * (w * (log(l) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.5e+239], N[Abs[N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[w, -1.55e+109], N[Not[LessEqual[w, 2.5]], $MachinePrecision]], N[(N[(l * N[(w * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(l + N[Abs[N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.50000000000000003e239

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.8%

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

      1. sub-neg 1.8%

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

      2. distribute-rgt-in 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r* 1.8%

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

   7. Applied egg-rr 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 21.5%

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

      3. pow2 21.5%

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

      4. fma-define 21.5%

         \[\leadsto \ell + \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \left(-\ell\right) \cdot w\right)\right)}}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{-\ell} \cdot \sqrt{-\ell}\right)} \cdot w\right)\right)}^{2}} \]

      6. sqrt-unprod 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot w\right)\right)}^{2}} \]

      7. sqr-neg 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \sqrt{\color{blue}{\ell \cdot \ell}} \cdot w\right)\right)}^{2}} \]

      8. sqrt-unprod 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot w\right)\right)}^{2}} \]

      9. add-sqr-sqrt 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\ell} \cdot w\right)\right)}^{2}} \]

   9. Applied egg-rr 79.9%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 79.9%

          \[\leadsto \ell + \sqrt{\color{blue}{\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right) \cdot \mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)}} \]

       2. rem-sqrt-square 60.6%

          \[\leadsto \ell + \color{blue}{\left|\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right|} \]

       3. fma-undefine 60.6%

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

       4. distribute-lft1-in 60.6%

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

       5. distribute-rgt1-in 60.6%

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

       6. *-lft-identity 60.6%

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

       7. distribute-rgt-out 60.6%

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

   11. Simplified 60.6%

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

   12. Taylor expanded in l around 0 64.6%

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

   if -2.50000000000000003e239 < w < -1.54999999999999996e109 or 2.5 < w

   1. Initial program 98.1%

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

      1. exp-neg 98.1%

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

      2. remove-double-neg 98.1%

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

      3. associate-*l/ 98.1%

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

      4. *-lft-identity 98.1%

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

      5. remove-double-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in w around 0 74.2%

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

      1. associate-*r* 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in w around inf 74.2%

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

   if -1.54999999999999996e109 < w < 2.5

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 83.8%

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

      1. add-sqr-sqrt 3.4%

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

      2. sqrt-unprod 12.8%

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

      3. pow2 12.8%

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

      4. sub-neg 12.8%

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

      5. distribute-rgt-in 9.4%

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

      6. neg-mul-1 9.4%

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

      7. *-commutative 9.4%

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

      8. distribute-rgt-in 12.8%

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

      9. distribute-lft-in 12.8%

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

      10. associate-*r* 12.8%

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

      11. *-commutative 12.8%

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

   7. Applied egg-rr 88.6%

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

      1. unpow2 12.8%

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

      2. rem-sqrt-square 8.9%

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

      3. associate-*r* 8.9%

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

      4. +-commutative 8.9%

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

   9. Simplified 87.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.5 \cdot 10^{+239}:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + 1\right)\right)\right|\\ \mathbf{elif}\;w \leq -1.55 \cdot 10^{+109} \lor \neg \left(w \leq 2.5\right):\\ \;\;\;\;\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \left|\ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.4 \cdot 10^{+237}:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + 1\right)\right)\right|\\ \mathbf{elif}\;w \leq -3.8 \cdot 10^{+19} \lor \neg \left(w \leq 1.25\right):\\ \;\;\;\;\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.4e+237)
   (fabs (* l (* w (+ (log l) 1.0))))
   (if (or (<= w -3.8e+19) (not (<= w 1.25)))
     (/ (* l (* w (log l))) (exp w))
     (+ l (* l (* w (+ (log l) -1.0)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.4e+237) {
		tmp = fabs((l * (w * (log(l) + 1.0))));
	} else if ((w <= -3.8e+19) || !(w <= 1.25)) {
		tmp = (l * (w * log(l))) / exp(w);
	} else {
		tmp = l + (l * (w * (log(l) + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.4d+237)) then
        tmp = abs((l * (w * (log(l) + 1.0d0))))
    else if ((w <= (-3.8d+19)) .or. (.not. (w <= 1.25d0))) then
        tmp = (l * (w * log(l))) / exp(w)
    else
        tmp = l + (l * (w * (log(l) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.4e+237) {
		tmp = Math.abs((l * (w * (Math.log(l) + 1.0))));
	} else if ((w <= -3.8e+19) || !(w <= 1.25)) {
		tmp = (l * (w * Math.log(l))) / Math.exp(w);
	} else {
		tmp = l + (l * (w * (Math.log(l) + -1.0)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.4e+237:
		tmp = math.fabs((l * (w * (math.log(l) + 1.0))))
	elif (w <= -3.8e+19) or not (w <= 1.25):
		tmp = (l * (w * math.log(l))) / math.exp(w)
	else:
		tmp = l + (l * (w * (math.log(l) + -1.0)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.4e+237)
		tmp = abs(Float64(l * Float64(w * Float64(log(l) + 1.0))));
	elseif ((w <= -3.8e+19) || !(w <= 1.25))
		tmp = Float64(Float64(l * Float64(w * log(l))) / exp(w));
	else
		tmp = Float64(l + Float64(l * Float64(w * Float64(log(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.4e+237)
		tmp = abs((l * (w * (log(l) + 1.0))));
	elseif ((w <= -3.8e+19) || ~((w <= 1.25)))
		tmp = (l * (w * log(l))) / exp(w);
	else
		tmp = l + (l * (w * (log(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.4e+237], N[Abs[N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[w, -3.8e+19], N[Not[LessEqual[w, 1.25]], $MachinePrecision]], N[(N[(l * N[(w * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(l + N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -1.39999999999999992e237

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.8%

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

      1. sub-neg 1.8%

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

      2. distribute-rgt-in 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r* 1.8%

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

   7. Applied egg-rr 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 21.5%

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

      3. pow2 21.5%

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

      4. fma-define 21.5%

         \[\leadsto \ell + \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \left(-\ell\right) \cdot w\right)\right)}}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{-\ell} \cdot \sqrt{-\ell}\right)} \cdot w\right)\right)}^{2}} \]

      6. sqrt-unprod 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot w\right)\right)}^{2}} \]

      7. sqr-neg 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \sqrt{\color{blue}{\ell \cdot \ell}} \cdot w\right)\right)}^{2}} \]

      8. sqrt-unprod 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot w\right)\right)}^{2}} \]

      9. add-sqr-sqrt 79.9%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\ell} \cdot w\right)\right)}^{2}} \]

   9. Applied egg-rr 79.9%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 79.9%

          \[\leadsto \ell + \sqrt{\color{blue}{\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right) \cdot \mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)}} \]

       2. rem-sqrt-square 60.6%

          \[\leadsto \ell + \color{blue}{\left|\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right|} \]

       3. fma-undefine 60.6%

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

       4. distribute-lft1-in 60.6%

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

       5. distribute-rgt1-in 60.6%

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

       6. *-lft-identity 60.6%

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

       7. distribute-rgt-out 60.6%

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

   11. Simplified 60.6%

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

   12. Taylor expanded in l around 0 64.6%

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

   if -1.39999999999999992e237 < w < -3.8e19 or 1.25 < w

   1. Initial program 98.7%

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

      1. exp-neg 98.7%

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

      2. remove-double-neg 98.7%

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

      3. associate-*l/ 98.7%

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

      4. *-lft-identity 98.7%

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

      5. remove-double-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in w around 0 59.0%

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

      1. associate-*r* 59.0%

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

   7. Simplified 59.0%

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

   8. Taylor expanded in w around inf 59.0%

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

   if -3.8e19 < w < 1.25

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 96.8%

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

   6. Taylor expanded in l around 0 96.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.4 \cdot 10^{+237}:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + 1\right)\right)\right|\\ \mathbf{elif}\;w \leq -3.8 \cdot 10^{+19} \lor \neg \left(w \leq 1.25\right):\\ \;\;\;\;\frac{\ell \cdot \left(w \cdot \log \ell\right)}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.0065:\\ \;\;\;\;\frac{\ell + \left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \left|\left(\ell \cdot w\right) \cdot \left(\log \ell + 1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.0065)
   (/ (+ l (* (* l w) (log l))) (exp w))
   (+ l (fabs (* (* l w) (+ (log l) 1.0))))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.0065) {
		tmp = (l + ((l * w) * log(l))) / exp(w);
	} else {
		tmp = l + fabs(((l * w) * (log(l) + 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 (l <= 0.0065d0) then
        tmp = (l + ((l * w) * log(l))) / exp(w)
    else
        tmp = l + abs(((l * w) * (log(l) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (l <= 0.0065) {
		tmp = (l + ((l * w) * Math.log(l))) / Math.exp(w);
	} else {
		tmp = l + Math.abs(((l * w) * (Math.log(l) + 1.0)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if l <= 0.0065:
		tmp = (l + ((l * w) * math.log(l))) / math.exp(w)
	else:
		tmp = l + math.fabs(((l * w) * (math.log(l) + 1.0)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.0065)
		tmp = Float64(Float64(l + Float64(Float64(l * w) * log(l))) / exp(w));
	else
		tmp = Float64(l + abs(Float64(Float64(l * w) * Float64(log(l) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (l <= 0.0065)
		tmp = (l + ((l * w) * log(l))) / exp(w);
	else
		tmp = l + abs(((l * w) * (log(l) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.0065], N[(N[(l + N[(N[(l * w), $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(l + N[Abs[N[(N[(l * w), $MachinePrecision] * N[(N[Log[l], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.0064999999999999997

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 98.5%

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

      1. associate-*r* 98.5%

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

   7. Simplified 98.5%

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

   if 0.0064999999999999997 < l

   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 59.1%

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

      1. sub-neg 59.1%

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

      2. distribute-rgt-in 59.1%

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

      3. *-commutative 59.1%

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

      4. associate-*r* 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. add-sqr-sqrt 30.4%

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

      2. sqrt-unprod 69.5%

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

      3. pow2 69.5%

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

      4. fma-define 69.5%

         \[\leadsto \ell + \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \left(-\ell\right) \cdot w\right)\right)}}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{-\ell} \cdot \sqrt{-\ell}\right)} \cdot w\right)\right)}^{2}} \]

      6. sqrt-unprod 69.8%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot w\right)\right)}^{2}} \]

      7. sqr-neg 69.8%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \sqrt{\color{blue}{\ell \cdot \ell}} \cdot w\right)\right)}^{2}} \]

      8. sqrt-unprod 88.4%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot w\right)\right)}^{2}} \]

      9. add-sqr-sqrt 88.4%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\ell} \cdot w\right)\right)}^{2}} \]

   9. Applied egg-rr 88.4%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 88.4%

          \[\leadsto \ell + \sqrt{\color{blue}{\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right) \cdot \mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)}} \]

       2. rem-sqrt-square 79.9%

          \[\leadsto \ell + \color{blue}{\left|\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right|} \]

       3. fma-undefine 79.9%

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

       4. distribute-lft1-in 79.9%

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

       5. distribute-rgt1-in 79.9%

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

       6. *-lft-identity 79.9%

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

       7. distribute-rgt-out 79.9%

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

   11. Simplified 79.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.0065:\\ \;\;\;\;\frac{\ell + \left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\ell + \left|\left(\ell \cdot w\right) \cdot \left(\log \ell + 1\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \mathbf{if}\;w \leq -0.0022:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\ell + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* l (* w (+ (log l) -1.0)))))
   (if (<= w -0.0022) (fabs t_0) (+ l t_0))))

C

double code(double w, double l) {
	double t_0 = l * (w * (log(l) + -1.0));
	double tmp;
	if (w <= -0.0022) {
		tmp = fabs(t_0);
	} else {
		tmp = l + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l * (w * (log(l) + (-1.0d0)))
    if (w <= (-0.0022d0)) then
        tmp = abs(t_0)
    else
        tmp = l + t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = l * (w * (Math.log(l) + -1.0));
	double tmp;
	if (w <= -0.0022) {
		tmp = Math.abs(t_0);
	} else {
		tmp = l + t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = l * (w * (math.log(l) + -1.0))
	tmp = 0
	if w <= -0.0022:
		tmp = math.fabs(t_0)
	else:
		tmp = l + t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(l * Float64(w * Float64(log(l) + -1.0)))
	tmp = 0.0
	if (w <= -0.0022)
		tmp = abs(t_0);
	else
		tmp = Float64(l + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = l * (w * (log(l) + -1.0));
	tmp = 0.0;
	if (w <= -0.0022)
		tmp = abs(t_0);
	else
		tmp = l + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -0.0022], N[Abs[t$95$0], $MachinePrecision], N[(l + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -0.00220000000000000013

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.6%

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

   6. Taylor expanded in w around inf 1.6%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 60.6%

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

      3. pow2 60.6%

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

      4. sub-neg 60.6%

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

      5. distribute-rgt-in 32.2%

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

      6. neg-mul-1 32.2%

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

      7. *-commutative 32.2%

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

      8. distribute-rgt-in 60.6%

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

      9. distribute-lft-in 60.6%

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

      10. associate-*r* 60.6%

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

      11. *-commutative 60.6%

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

   8. Applied egg-rr 60.6%

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

      1. unpow2 60.6%

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

      2. rem-sqrt-square 33.0%

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

      3. associate-*r* 34.2%

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

      4. +-commutative 34.2%

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

   10. Simplified 34.2%

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

   if -0.00220000000000000013 < w

   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 85.6%

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

   6. Taylor expanded in l around 0 85.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.0022:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.0032:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + 1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.0032)
   (fabs (* l (* w (+ (log l) 1.0))))
   (+ l (* l (* w (+ (log l) -1.0))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.0032) {
		tmp = fabs((l * (w * (log(l) + 1.0))));
	} else {
		tmp = l + (l * (w * (log(l) + -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 <= (-0.0032d0)) then
        tmp = abs((l * (w * (log(l) + 1.0d0))))
    else
        tmp = l + (l * (w * (log(l) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.0032) {
		tmp = Math.abs((l * (w * (Math.log(l) + 1.0))));
	} else {
		tmp = l + (l * (w * (Math.log(l) + -1.0)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.0032:
		tmp = math.fabs((l * (w * (math.log(l) + 1.0))))
	else:
		tmp = l + (l * (w * (math.log(l) + -1.0)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.0032)
		tmp = abs(Float64(l * Float64(w * Float64(log(l) + 1.0))));
	else
		tmp = Float64(l + Float64(l * Float64(w * Float64(log(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.0032)
		tmp = abs((l * (w * (log(l) + 1.0))));
	else
		tmp = l + (l * (w * (log(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.0032], N[Abs[N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l + N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.00320000000000000015

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.6%

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

      1. sub-neg 1.6%

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

      2. distribute-rgt-in 1.6%

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

      3. *-commutative 1.6%

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

      4. associate-*r* 1.6%

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

   7. Applied egg-rr 1.6%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 32.2%

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

      3. pow2 32.2%

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

      4. fma-define 32.2%

         \[\leadsto \ell + \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \left(-\ell\right) \cdot w\right)\right)}}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{-\ell} \cdot \sqrt{-\ell}\right)} \cdot w\right)\right)}^{2}} \]

      6. sqrt-unprod 60.6%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot w\right)\right)}^{2}} \]

      7. sqr-neg 60.6%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \sqrt{\color{blue}{\ell \cdot \ell}} \cdot w\right)\right)}^{2}} \]

      8. sqrt-unprod 60.6%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot w\right)\right)}^{2}} \]

      9. add-sqr-sqrt 60.6%

         \[\leadsto \ell + \sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \color{blue}{\ell} \cdot w\right)\right)}^{2}} \]

   9. Applied egg-rr 60.6%

      \[\leadsto \ell + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 60.6%

          \[\leadsto \ell + \sqrt{\color{blue}{\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right) \cdot \mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)}} \]

       2. rem-sqrt-square 33.0%

          \[\leadsto \ell + \color{blue}{\left|\mathsf{fma}\left(\log \ell, \ell \cdot w, \ell \cdot w\right)\right|} \]

       3. fma-undefine 33.0%

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

       4. distribute-lft1-in 33.0%

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

       5. distribute-rgt1-in 33.0%

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

       6. *-lft-identity 33.0%

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

       7. distribute-rgt-out 33.0%

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

   11. Simplified 33.0%

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

   12. Taylor expanded in l around 0 34.2%

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

   if -0.00320000000000000015 < w

   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 85.6%

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

   6. Taylor expanded in l around 0 85.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.0032:\\ \;\;\;\;\left|\ell \cdot \left(w \cdot \left(\log \ell + 1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.00335:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.00335) (* l (- w)) (+ l (* l (* w (+ (log l) -1.0))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.00335) {
		tmp = l * -w;
	} else {
		tmp = l + (l * (w * (log(l) + -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 <= (-0.00335d0)) then
        tmp = l * -w
    else
        tmp = l + (l * (w * (log(l) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.00335) {
		tmp = l * -w;
	} else {
		tmp = l + (l * (w * (Math.log(l) + -1.0)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.00335:
		tmp = l * -w
	else:
		tmp = l + (l * (w * (math.log(l) + -1.0)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.00335)
		tmp = Float64(l * Float64(-w));
	else
		tmp = Float64(l + Float64(l * Float64(w * Float64(log(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.00335)
		tmp = l * -w;
	else
		tmp = l + (l * (w * (log(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.00335], N[(l * (-w)), $MachinePrecision], N[(l + N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.00335000000000000011

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.6%

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

   6. Taylor expanded in w around inf 1.6%

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

      1. add-log-exp 1.5%

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

      2. *-commutative 1.5%

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

      3. exp-to-pow 1.5%

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

   8. Applied egg-rr 1.5%

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

   9. Taylor expanded in l around inf 32.0%

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

       1. neg-mul-1 32.0%

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

   11. Simplified 32.0%

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

   if -0.00335000000000000011 < w

   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 85.6%

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

   6. Taylor expanded in l around 0 85.6%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.00335:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.09:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.09) (* l (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.09) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.09d0)) then
        tmp = l * -w
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.09) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.09:
		tmp = l * -w
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.09)
		tmp = Float64(l * Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.09)
		tmp = l * -w;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.09], N[(l * (-w)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.089999999999999997

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 1.6%

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

   6. Taylor expanded in w around inf 1.6%

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

      1. add-log-exp 1.5%

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

      2. *-commutative 1.5%

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

      3. exp-to-pow 1.5%

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

   8. Applied egg-rr 1.5%

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

   9. Taylor expanded in l around inf 32.0%

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

       1. neg-mul-1 32.0%

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

   11. Simplified 32.0%

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

   if -0.089999999999999997 < w

   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 85.6%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.09:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in w around 0 59.8%

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

6. Final simplification 59.8%

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

Reproduce

herbie shell --seed 2024053 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))