exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 16.7s

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 3: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 580\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 580.0)))
   (exp (- w))
   (- l (* w (+ l (* w (* l -0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 580.0)) {
		tmp = exp(-w);
	} else {
		tmp = l - (w * (l + (w * (l * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.7d0)) .or. (.not. (w <= 580.0d0))) then
        tmp = exp(-w)
    else
        tmp = l - (w * (l + (w * (l * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 580.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l - (w * (l + (w * (l * -0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 580.0):
		tmp = math.exp(-w)
	else:
		tmp = l - (w * (l + (w * (l * -0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 580.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(l * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 580.0)))
		tmp = exp(-w);
	else
		tmp = l - (w * (l + (w * (l * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 580.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l - N[(w * N[(l + N[(w * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 580 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 99.9%

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

      5. exp-1-e 99.9%

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

      6. log-div 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. fma-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in w around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in w around inf 100.0%

       \[\leadsto \color{blue}{e^{-1 \cdot \left(w \cdot \log e\right)}} \]
   11. Step-by-step derivation

       1. exp-prod 99.9%

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

       2. log-E 99.9%

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

       3. *-rgt-identity 99.9%

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

       4. exp-prod 100.0%

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

       5. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

   if -0.69999999999999996 < w < 580

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 96.4%

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

   6. Taylor expanded in w around 0 96.4%

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

      1. associate-*r* 96.4%

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

      2. mul-1-neg 96.4%

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

      3. distribute-rgt-out 96.4%

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

      4. metadata-eval 96.4%

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

   8. Simplified 96.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 580\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

Alternative 5: 75.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(w \cdot \left(\left(\ell - \ell \cdot 0.5\right) + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right) - \ell\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (+
  l
  (*
   w
   (-
    (*
     w
     (+
      (- l (* l 0.5))
      (* w (- (- (* l 0.5) l) (+ (* l -0.5) (* l 0.16666666666666666))))))
    l))))

C

double code(double w, double l) {
	return l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * ((w * ((l - (l * 0.5d0)) + (w * (((l * 0.5d0) - l) - ((l * (-0.5d0)) + (l * 0.16666666666666666d0)))))) - l))
end function

Java

public static double code(double w, double l) {
	return l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l));
}

Python

def code(w, l):
	return l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(Float64(w * Float64(Float64(l - Float64(l * 0.5)) + Float64(w * Float64(Float64(Float64(l * 0.5) - l) - Float64(Float64(l * -0.5) + Float64(l * 0.16666666666666666)))))) - l)))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * ((w * ((l - (l * 0.5)) + (w * (((l * 0.5) - l) - ((l * -0.5) + (l * 0.16666666666666666)))))) - l));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

6. Taylor expanded in w around 0 76.8%

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

7. Final simplification 76.8%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(\left(\ell - \ell \cdot 0.5\right) + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right) - \ell\right) \]
8. Add Preprocessing

Alternative 6: 73.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.9e+146)
   (+ 1.0 (* w (+ -1.0 (* w 0.5))))
   (+ l (* w (* l (* w 0.5))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.9e+146) {
		tmp = 1.0 + (w * (-1.0 + (w * 0.5)));
	} else {
		tmp = l + (w * (l * (w * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.9d+146)) then
        tmp = 1.0d0 + (w * ((-1.0d0) + (w * 0.5d0)))
    else
        tmp = l + (w * (l * (w * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.9e+146) {
		tmp = 1.0 + (w * (-1.0 + (w * 0.5)));
	} else {
		tmp = l + (w * (l * (w * 0.5)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.9e+146:
		tmp = 1.0 + (w * (-1.0 + (w * 0.5)))
	else:
		tmp = l + (w * (l * (w * 0.5)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.9e+146)
		tmp = Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * 0.5))));
	else
		tmp = Float64(l + Float64(w * Float64(l * Float64(w * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.9e+146)
		tmp = 1.0 + (w * (-1.0 + (w * 0.5)));
	else
		tmp = l + (w * (l * (w * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.9e+146], N[(1.0 + N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l + N[(w * N[(l * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.8999999999999999e146

   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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around 0 95.8%

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

       1. log-E 95.8%

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

       2. metadata-eval 95.8%

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

       3. associate-*r* 95.8%

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

       4. log-E 95.8%

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

       5. metadata-eval 95.8%

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

       6. associate-*r* 95.8%

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

   12. Simplified 95.8%

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

   if -1.8999999999999999e146 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 97.1%

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

   6. Taylor expanded in w around 0 70.5%

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

      1. associate-*r* 70.5%

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

      2. mul-1-neg 70.5%

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

      3. distribute-rgt-out 70.5%

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

      4. metadata-eval 70.5%

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

   8. Simplified 70.5%

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

   9. Taylor expanded in w around inf 70.5%

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

       1. *-commutative 70.5%

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

       2. associate-*l* 70.5%

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

   11. Simplified 70.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* l (* w (+ -1.0 (* w 0.5))))))

C

double code(double w, double l) {
	return l + (l * (w * (-1.0 + (w * 0.5))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (l * (w * ((-1.0d0) + (w * 0.5d0))))
end function

Java

public static double code(double w, double l) {
	return l + (l * (w * (-1.0 + (w * 0.5))));
}

Python

def code(w, l):
	return l + (l * (w * (-1.0 + (w * 0.5))))

Julia

function code(w, l)
	return Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * 0.5)))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (l * (w * (-1.0 + (w * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

6. Taylor expanded in w around 0 71.3%

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

   1. associate-*r* 71.3%

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

   2. mul-1-neg 71.3%

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

   3. distribute-rgt-out 71.3%

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

   4. metadata-eval 71.3%

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

8. Simplified 71.3%

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

9. Taylor expanded in l around 0 74.6%

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

10. Final simplification 74.6%

    \[\leadsto \ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right) \]
11. Add Preprocessing

Alternative 8: 57.7% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;1 - w\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.7) (- 1.0 w) (+ l (* w l))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = 1.0 - w;
	} else {
		tmp = 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 <= (-0.7d0)) then
        tmp = 1.0d0 - w
    else
        tmp = l + (w * l)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = 1.0 - w;
	} else {
		tmp = l + (w * l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = 1.0 - w
	else:
		tmp = l + (w * l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = Float64(1.0 - w);
	else
		tmp = Float64(l + Float64(w * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.7)
		tmp = 1.0 - w;
	else
		tmp = l + (w * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[(1.0 - w), $MachinePrecision], N[(l + N[(w * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996

   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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 99.9%

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

      5. exp-1-e 99.9%

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

      6. log-div 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. fma-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in w around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in w around 0 5.5%

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

       1. mul-1-neg 5.5%

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

       2. log-E 5.5%

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

       3. metadata-eval 5.5%

          \[\leadsto 1 + \left(-w \cdot \color{blue}{{1}^{3}}\right) \]

       4. log-E 5.5%

          \[\leadsto 1 + \left(-w \cdot {\color{blue}{\log e}}^{3}\right) \]

       5. unsub-neg 5.5%

          \[\leadsto \color{blue}{1 - w \cdot {\log e}^{3}} \]

       6. log-E 5.5%

          \[\leadsto 1 - w \cdot {\color{blue}{1}}^{3} \]

       7. metadata-eval 5.5%

          \[\leadsto 1 - w \cdot \color{blue}{1} \]

       8. *-rgt-identity 5.5%

          \[\leadsto 1 - \color{blue}{w} \]

   12. Simplified 5.5%

       \[\leadsto \color{blue}{1 - w} \]

   if -0.69999999999999996 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 97.0%

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

      1. add-log-exp 25.1%

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

      2. div-inv 25.1%

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

      3. rec-exp 25.1%

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

      4. add-sqr-sqrt 3.8%

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

      5. sqrt-unprod 8.2%

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

      6. sqr-neg 8.2%

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

      7. sqrt-unprod 4.4%

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

      8. add-sqr-sqrt 8.2%

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

   7. Applied egg-rr 8.2%

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

   8. Taylor expanded in w around 0 80.5%

      \[\leadsto \color{blue}{\ell + \ell \cdot w} \]
   9. Step-by-step derivation

      1. *-commutative 80.5%

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

   10. Simplified 80.5%

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

Alternative 9: 70.5% accurate, 33.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

6. Taylor expanded in w around 0 71.3%

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

   1. associate-*r* 71.3%

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

   2. mul-1-neg 71.3%

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

   3. distribute-rgt-out 71.3%

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

   4. metadata-eval 71.3%

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

8. Simplified 71.3%

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

9. Taylor expanded in w around inf 71.3%

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

    1. *-commutative 71.3%

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

    2. associate-*l* 71.3%

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

11. Simplified 71.3%

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

Alternative 10: 64.1% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

6. Taylor expanded in w around 0 63.9%

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

   1. mul-1-neg 63.9%

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

   2. *-commutative 63.9%

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

   3. unsub-neg 63.9%

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

   4. *-commutative 63.9%

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

8. Simplified 63.9%

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

9. Final simplification 63.9%

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

Alternative 11: 4.9% accurate, 101.7× speedup

Math

\[\begin{array}{l} \\ 1 - w \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (- 1.0 w))

C

double code(double w, double l) {
	return 1.0 - w;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 1.0d0 - w
end function

Java

public static double code(double w, double l) {
	return 1.0 - w;
}

Python

def code(w, l):
	return 1.0 - w

Julia

function code(w, l)
	return Float64(1.0 - w)
end

MATLAB

function tmp = code(w, l)
	tmp = 1.0 - w;
end

Wolfram

code[w_, l_] := N[(1.0 - w), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 95.2%

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

   2. pow1 95.2%

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

   3. log-pow 95.2%

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

   4. exp-prod 94.6%

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

   5. exp-1-e 94.6%

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

   6. log-div 94.6%

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

   7. log-pow 94.6%

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

   8. add-log-exp 94.6%

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

   9. fma-neg 94.6%

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

6. Applied egg-rr 94.6%

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

7. Taylor expanded in w around inf 46.5%

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

   1. mul-1-neg 46.5%

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

9. Simplified 46.5%

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

10. Taylor expanded in w around 0 5.1%

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

    1. mul-1-neg 5.1%

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

    2. log-E 5.1%

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

    3. metadata-eval 5.1%

       \[\leadsto 1 + \left(-w \cdot \color{blue}{{1}^{3}}\right) \]

    4. log-E 5.1%

       \[\leadsto 1 + \left(-w \cdot {\color{blue}{\log e}}^{3}\right) \]

    5. unsub-neg 5.1%

       \[\leadsto \color{blue}{1 - w \cdot {\log e}^{3}} \]

    6. log-E 5.1%

       \[\leadsto 1 - w \cdot {\color{blue}{1}}^{3} \]

    7. metadata-eval 5.1%

       \[\leadsto 1 - w \cdot \color{blue}{1} \]

    8. *-rgt-identity 5.1%

       \[\leadsto 1 - \color{blue}{w} \]

12. Simplified 5.1%

    \[\leadsto \color{blue}{1 - w} \]
13. Add Preprocessing

Alternative 12: 4.4% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 1.0)

C

double code(double w, double l) {
	return 1.0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 1.0d0
end function

Java

public static double code(double w, double l) {
	return 1.0;
}

Python

def code(w, l):
	return 1.0

Julia

function code(w, l)
	return 1.0
end

MATLAB

function tmp = code(w, l)
	tmp = 1.0;
end

Wolfram

code[w_, l_] := 1.0

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 95.2%

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

   2. pow1 95.2%

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

   3. log-pow 95.2%

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

   4. exp-prod 94.6%

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

   5. exp-1-e 94.6%

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

   6. log-div 94.6%

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

   7. log-pow 94.6%

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

   8. add-log-exp 94.6%

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

   9. fma-neg 94.6%

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

6. Applied egg-rr 94.6%

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

7. Taylor expanded in w around inf 46.5%

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

   1. mul-1-neg 46.5%

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

9. Simplified 46.5%

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

10. Taylor expanded in w around 0 4.5%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))