exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 31.5s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. w = -2.00000000000000007e154
  2. l = 0.0

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

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

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 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 14.2\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \ell \cdot w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 14.2))) (exp (- w)) (- l (* l w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 14.2)) {
		tmp = exp(-w);
	} else {
		tmp = l - (l * w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.7d0)) .or. (.not. (w <= 14.2d0))) then
        tmp = exp(-w)
    else
        tmp = l - (l * w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 14.2)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l - (l * w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 14.2):
		tmp = math.exp(-w)
	else:
		tmp = l - (l * w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 14.2))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l - Float64(l * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 14.2)))
		tmp = exp(-w);
	else
		tmp = l - (l * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 14.2]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 14.199999999999999 < w

   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 l around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. log-rec 99.2%

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

      4. remove-double-neg 99.2%

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

      5. +-rgt-identity 99.2%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

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

      7. +-rgt-identity 100.0%

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

      8. *-commutative 100.0%

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

      9. remove-double-neg 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. distribute-lft-neg-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 99.2%

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

      1. neg-mul-1 99.2%

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

   10. Simplified 99.2%

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

   if -0.69999999999999996 < w < 14.199999999999999

   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 95.3%

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

   6. Taylor expanded in w around 0 95.3%

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

      1. mul-1-neg 95.3%

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

      2. unsub-neg 95.3%

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

      3. *-commutative 95.3%

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

   8. Simplified 95.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 14.2\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \ell \cdot w\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in w around 0 97.1%

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

6. Final simplification 97.1%

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

Alternative 4: 75.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot 0.5 - \ell\\ \ell - w \cdot \left(\ell + w \cdot \left(t\_0 - w \cdot \left(t\_0 - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- (* l 0.5) l)))
   (-
    l
    (*
     w
     (+
      l
      (* w (- t_0 (* w (- t_0 (+ (* l -0.5) (* l 0.16666666666666666)))))))))))

C

double code(double w, double l) {
	double t_0 = (l * 0.5) - l;
	return l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = (l * 0.5d0) - l
    code = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * (-0.5d0)) + (l * 0.16666666666666666d0))))))))
end function

Java

public static double code(double w, double l) {
	double t_0 = (l * 0.5) - l;
	return l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))));
}

Python

def code(w, l):
	t_0 = (l * 0.5) - l
	return l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))))

Julia

function code(w, l)
	t_0 = Float64(Float64(l * 0.5) - l)
	return Float64(l - Float64(w * Float64(l + Float64(w * Float64(t_0 - Float64(w * Float64(t_0 - Float64(Float64(l * -0.5) + Float64(l * 0.16666666666666666)))))))))
end

MATLAB

function tmp = code(w, l)
	t_0 = (l * 0.5) - l;
	tmp = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))));
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision]}, N[(l - N[(w * N[(l + N[(w * N[(t$95$0 - N[(w * N[(t$95$0 - N[(N[(l * -0.5), $MachinePrecision] + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
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 77.8%

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

7. Final simplification 77.8%

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

Alternative 5: 74.2% 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 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
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 72.0%

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

   1. associate-*r* 72.0%

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

   2. mul-1-neg 72.0%

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

   3. distribute-rgt-out 72.0%

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

   4. metadata-eval 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in l around 0 76.7%

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

10. Final simplification 76.7%

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

Alternative 6: 70.5% accurate, 33.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
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 72.0%

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

   1. associate-*r* 72.0%

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

   2. mul-1-neg 72.0%

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

   3. distribute-rgt-out 72.0%

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

   4. metadata-eval 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in w around inf 72.0%

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

    1. associate-*r* 72.0%

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

    2. *-commutative 72.0%

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

11. Simplified 72.0%

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

12. Final simplification 72.0%

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

Alternative 7: 63.7% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
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 61.5%

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

   1. mul-1-neg 61.5%

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

   2. unsub-neg 61.5%

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

   3. *-commutative 61.5%

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

8. Simplified 61.5%

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

9. Final simplification 61.5%

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

Alternative 8: 56.9% 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 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in w around 0 51.4%

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

   1. +-commutative 51.4%

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

   2. fma-define 51.4%

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

   3. sub-neg 51.4%

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

   4. *-commutative 51.4%

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

   5. neg-mul-1 51.4%

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

   6. distribute-rgt-out 51.4%

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

7. Simplified 51.4%

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

8. Taylor expanded in l around 0 52.1%

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

9. Taylor expanded in w around 0 52.2%

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

10. Final simplification 52.2%

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

Reproduce

herbie shell --seed 2024060 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))