exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 13.1s

Alternatives: 9

Speedup: N/A×

• 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: 99.4% 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 99.1%

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

   1. exp-neg 99.1%

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

   2. remove-double-neg 99.1%

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

   3. associate-*l/ 99.1%

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

   4. *-lft-identity 99.1%

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

   5. remove-double-neg 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.66 \lor \neg \left(w \leq 1500\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.66) (not (<= w 1500.0))) (exp (- w)) (* l (exp w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.66) || !(w <= 1500.0)) {
		tmp = exp(-w);
	} else {
		tmp = l * exp(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 <= (-0.66d0)) .or. (.not. (w <= 1500.0d0))) then
        tmp = exp(-w)
    else
        tmp = l * exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.66) || !(w <= 1500.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l * Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.66) or not (w <= 1500.0):
		tmp = math.exp(-w)
	else:
		tmp = l * math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.66) || !(w <= 1500.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l * exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.66) || ~((w <= 1500.0)))
		tmp = exp(-w);
	else
		tmp = l * exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.66], N[Not[LessEqual[w, 1500.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l * N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.660000000000000031 or 1500 < 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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-div 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 100.0%

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

      1. div-exp 100.0%

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

      2. mul-1-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. log-rec 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around inf 100.0%

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

       1. neg-mul-1 100.0%

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

   13. Simplified 100.0%

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

   if -0.660000000000000031 < w < 1500

   1. Initial program 98.5%

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

   3. Taylor expanded in w around 0 93.8%

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

      1. pow1 93.8%

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

      2. *-commutative 93.8%

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

      3. add-sqr-sqrt 37.8%

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

      4. sqrt-unprod 94.4%

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

      5. sqr-neg 94.4%

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

      6. sqrt-unprod 56.6%

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

      7. add-sqr-sqrt 94.4%

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

   5. Applied egg-rr 94.4%

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

      1. unpow1 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.66 \lor \neg \left(w \leq 1500\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 1500\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 1500.0)))
   (exp (- w))
   (+ l (* w (* l (+ -1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 1500.0)) {
		tmp = exp(-w);
	} else {
		tmp = l + (w * (l * (-1.0 + (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 <= (-0.7d0)) .or. (.not. (w <= 1500.0d0))) then
        tmp = exp(-w)
    else
        tmp = l + (w * (l * ((-1.0d0) + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 1500.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 1500.0):
		tmp = math.exp(-w)
	else:
		tmp = l + (w * (l * (-1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 1500.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l + Float64(w * Float64(l * Float64(-1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 1500.0)))
		tmp = exp(-w);
	else
		tmp = l + (w * (l * (-1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 1500.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l + N[(w * N[(l * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 1500 < 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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-div 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 100.0%

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

      1. div-exp 100.0%

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

      2. mul-1-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. log-rec 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around inf 100.0%

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

       1. neg-mul-1 100.0%

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

   13. Simplified 100.0%

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

   if -0.69999999999999996 < w < 1500

   1. Initial program 98.5%

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

   3. Taylor expanded in w around 0 93.8%

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

   4. Taylor expanded in w around 0 93.7%

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

   5. Taylor expanded in w around 0 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*r* 93.8%

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

      3. *-commutative 93.8%

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

      4. associate-*l* 93.8%

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

      5. distribute-lft-out 93.8%

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

      6. *-commutative 93.8%

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

   7. Simplified 93.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 1500\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 2.9× speedup

Math

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

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(Float64(-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 99.1%

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

3. Taylor expanded in w around 0 96.4%

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

4. Final simplification 96.4%

   \[\leadsto \ell \cdot e^{-w} \]
5. Add Preprocessing

Alternative 5: 97.3% 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 99.1%

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

   1. exp-neg 99.1%

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

   2. remove-double-neg 99.1%

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

   3. associate-*l/ 99.1%

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

   4. *-lft-identity 99.1%

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

   5. remove-double-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in w around 0 96.4%

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

6. Final simplification 96.4%

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

Alternative 6: 77.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (* l (+ 1.0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666))))))))

C

double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 + (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0)))))))
end function

Java

public static double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
}

Python

def code(w, l):
	return l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
end

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in w around 0 96.4%

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

4. Taylor expanded in w around 0 74.9%

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

5. Taylor expanded in l around 0 76.4%

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

6. Final simplification 76.4%

   \[\leadsto \ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \]
7. Add Preprocessing

Alternative 7: 70.5% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in w around 0 96.4%

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

4. Taylor expanded in w around 0 74.9%

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

5. Taylor expanded in w around 0 70.2%

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

   1. *-commutative 70.2%

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

   2. associate-*r* 70.2%

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

   3. *-commutative 70.2%

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

   4. associate-*l* 70.2%

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

   5. distribute-lft-out 70.2%

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

   6. *-commutative 70.2%

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

7. Simplified 70.2%

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

8. Final simplification 70.2%

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

Alternative 8: 63.8% 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 99.1%

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

   1. exp-neg 99.1%

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

   2. remove-double-neg 99.1%

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

   3. associate-*l/ 99.1%

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

   4. *-lft-identity 99.1%

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

   5. remove-double-neg 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
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 63.5%

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

   1. mul-1-neg 63.5%

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

   2. unsub-neg 63.5%

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

8. Simplified 63.5%

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

9. Final simplification 63.5%

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

Alternative 9: 56.7% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in w around 0 96.4%

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

4. Taylor expanded in w around 0 55.3%

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

5. Final simplification 55.3%

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

Reproduce

herbie shell --seed 2024057 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))