exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 13.7s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\ell}^{\left({\left({\left(e^{w}\right)}^{2}\right)}^{0.3333333333333333}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.3333333333333333}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (pow
   (pow l (pow (pow (exp w) 2.0) 0.3333333333333333))
   (pow (exp w) 0.3333333333333333))
  (exp w)))

C

double code(double w, double l) {
	return pow(pow(l, pow(pow(exp(w), 2.0), 0.3333333333333333)), pow(exp(w), 0.3333333333333333)) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l ** ((exp(w) ** 2.0d0) ** 0.3333333333333333d0)) ** (exp(w) ** 0.3333333333333333d0)) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow(Math.pow(l, Math.pow(Math.pow(Math.exp(w), 2.0), 0.3333333333333333)), Math.pow(Math.exp(w), 0.3333333333333333)) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(math.pow(l, math.pow(math.pow(math.exp(w), 2.0), 0.3333333333333333)), math.pow(math.exp(w), 0.3333333333333333)) / math.exp(w)

Julia

function code(w, l)
	return Float64(((l ^ ((exp(w) ^ 2.0) ^ 0.3333333333333333)) ^ (exp(w) ^ 0.3333333333333333)) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l ^ ((exp(w) ^ 2.0) ^ 0.3333333333333333)) ^ (exp(w) ^ 0.3333333333333333)) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[N[Power[l, N[Power[N[Power[N[Exp[w], $MachinePrecision], 2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision], N[Power[N[Exp[w], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.3%

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

   2. unpow-prod-down 99.3%

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

5. Applied egg-rr 99.3%

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

   1. pow-prod-down 99.3%

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

   2. add-sqr-sqrt 99.7%

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

   3. add-cube-cbrt 99.6%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}\right)}}}{e^{w}} \]

   4. pow-unpow 99.6%

      \[\leadsto \frac{\color{blue}{{\left({\ell}^{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right)}\right)}^{\left(\sqrt[3]{e^{w}}\right)}}}{e^{w}} \]

   5. pow2 99.6%

      \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left({\left(\sqrt[3]{e^{w}}\right)}^{2}\right)}}\right)}^{\left(\sqrt[3]{e^{w}}\right)}}{e^{w}} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{\color{blue}{{\left({\ell}^{\left({\left(\sqrt[3]{e^{w}}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{e^{w}}\right)}}}{e^{w}} \]

8. Taylor expanded in w around inf 99.7%

   \[\leadsto \frac{\color{blue}{{\left({\ell}^{\left({\left({\left(e^{w}\right)}^{2}\right)}^{0.3333333333333333}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.3333333333333333}\right)}}}{e^{w}} \]

9. Final simplification 99.7%

   \[\leadsto \frac{{\left({\ell}^{\left({\left({\left(e^{w}\right)}^{2}\right)}^{0.3333333333333333}\right)}\right)}^{\left({\left(e^{w}\right)}^{0.3333333333333333}\right)}}{e^{w}} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 3: 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 97.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 205\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \ell \cdot w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 205.0))) (exp (- w)) (- l (* l w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 205.0)) {
		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 <= 205.0d0))) 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 <= 205.0)) {
		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 <= 205.0):
		tmp = math.exp(-w)
	else:
		tmp = l - (l * w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 205.0))
		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 <= 205.0)))
		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, 205.0]], $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 205 < 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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. unpow-prod-down 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. pow-prod-down 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. exp-to-pow 100.0%

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

      4. div-exp 100.0%

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

      5. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in w around inf 99.8%

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

      1. neg-mul-1 99.8%

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

   10. Simplified 99.8%

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

   if -0.69999999999999996 < w < 205

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. associate-*l/ 99.4%

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

      3. *-lft-identity 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in w around 0 96.4%

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

   5. Taylor expanded in w around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. unsub-neg 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 205\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \ell \cdot w\\ \end{array} \]

Alternative 5: 97.8% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 97.7%

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

5. Final simplification 97.7%

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

Alternative 6: 67.0% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(\ell \cdot w\right) \cdot \left(\ell \cdot w\right)}{\ell + \ell \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \ell \cdot w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 5e-118)
   (* l (- 1.0 w))
   (if (<= l 7e+72)
     (/ (- (* l l) (* (* l w) (* l w))) (+ l (* l w)))
     (- l (* l w)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 5e-118) {
		tmp = l * (1.0 - w);
	} else if (l <= 7e+72) {
		tmp = ((l * l) - ((l * w) * (l * w))) / (l + (l * 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 (l <= 5d-118) then
        tmp = l * (1.0d0 - w)
    else if (l <= 7d+72) then
        tmp = ((l * l) - ((l * w) * (l * w))) / (l + (l * w))
    else
        tmp = l - (l * w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (l <= 5e-118) {
		tmp = l * (1.0 - w);
	} else if (l <= 7e+72) {
		tmp = ((l * l) - ((l * w) * (l * w))) / (l + (l * w));
	} else {
		tmp = l - (l * w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if l <= 5e-118:
		tmp = l * (1.0 - w)
	elif l <= 7e+72:
		tmp = ((l * l) - ((l * w) * (l * w))) / (l + (l * w))
	else:
		tmp = l - (l * w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 5e-118)
		tmp = Float64(l * Float64(1.0 - w));
	elseif (l <= 7e+72)
		tmp = Float64(Float64(Float64(l * l) - Float64(Float64(l * w) * Float64(l * w))) / Float64(l + Float64(l * w)));
	else
		tmp = Float64(l - Float64(l * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (l <= 5e-118)
		tmp = l * (1.0 - w);
	elseif (l <= 7e+72)
		tmp = ((l * l) - ((l * w) * (l * w))) / (l + (l * w));
	else
		tmp = l - (l * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[l, 5e-118], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7e+72], N[(N[(N[(l * l), $MachinePrecision] - N[(N[(l * w), $MachinePrecision] * N[(l * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l + N[(l * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.00000000000000015e-118

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in w around 0 97.5%

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

   5. Taylor expanded in w around 0 45.5%

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

      1. +-commutative 45.5%

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

      2. mul-1-neg 45.5%

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

      3. unsub-neg 45.5%

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

   7. Simplified 45.5%

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

   8. Taylor expanded in l around 0 45.5%

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

   if 5.00000000000000015e-118 < l < 7.0000000000000002e72

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 100.0%

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

   5. Taylor expanded in w around 0 49.8%

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

      1. +-commutative 49.8%

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

      2. mul-1-neg 49.8%

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

      3. unsub-neg 49.8%

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

   7. Simplified 49.8%

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

      1. add-exp-log 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. add-exp-log 49.8%

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

       2. sub-neg 49.8%

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

       3. flip-+ 69.9%

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

       4. distribute-rgt-neg-in 69.9%

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

       5. distribute-rgt-neg-in 69.9%

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

       6. distribute-rgt-neg-in 69.9%

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

   11. Applied egg-rr 69.9%

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

   if 7.0000000000000002e72 < l

   1. Initial program 99.6%

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

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in w around 0 95.9%

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

   5. Taylor expanded in w around 0 86.8%

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

      1. +-commutative 86.8%

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

      2. mul-1-neg 86.8%

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

      3. unsub-neg 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(\ell \cdot w\right) \cdot \left(\ell \cdot w\right)}{\ell + \ell \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \ell \cdot w\\ \end{array} \]

Alternative 7: 64.1% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 97.7%

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

5. Taylor expanded in w around 0 61.4%

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

   1. +-commutative 61.4%

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

   2. mul-1-neg 61.4%

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

   3. unsub-neg 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in l around 0 61.4%

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

9. Final simplification 61.4%

   \[\leadsto \ell \cdot \left(1 - w\right) \]

Alternative 8: 64.1% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 97.7%

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

5. Taylor expanded in w around 0 61.4%

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

   1. +-commutative 61.4%

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

   2. mul-1-neg 61.4%

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

   3. unsub-neg 61.4%

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

7. Simplified 61.4%

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

8. Final simplification 61.4%

   \[\leadsto \ell - \ell \cdot w \]

Alternative 9: 11.3% accurate, 76.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 97.7%

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

5. Taylor expanded in w around 0 61.4%

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

   1. +-commutative 61.4%

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

   2. mul-1-neg 61.4%

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

   3. unsub-neg 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in w around inf 11.0%

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

   1. neg-mul-1 11.0%

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

   2. distribute-rgt-neg-in 11.0%

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

10. Simplified 11.0%

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

11. Final simplification 11.0%

    \[\leadsto \ell \cdot \left(-w\right) \]

Reproduce

herbie shell --seed 2023199 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))