exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 11.4s

Alternatives: 8

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: 99.5% 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: 99.5% 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 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 99.5% 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 99.2%

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

   1. exp-neg 99.2%

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

   2. associate-*l/ 99.2%

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

   3. *-lft-identity 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 97.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.69 \lor \neg \left(w \leq 24.5\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - w \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.69) (not (<= w 24.5))) (exp (- w)) (- l (* w l))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.69) || !(w <= 24.5)) {
		tmp = exp(-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 <= (-0.69d0)) .or. (.not. (w <= 24.5d0))) then
        tmp = exp(-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 <= -0.69) || !(w <= 24.5)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l - (w * l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.69) or not (w <= 24.5):
		tmp = math.exp(-w)
	else:
		tmp = l - (w * l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.69) || !(w <= 24.5))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l - Float64(w * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.69) || ~((w <= 24.5)))
		tmp = exp(-w);
	else
		tmp = l - (w * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.69], N[Not[LessEqual[w, 24.5]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.68999999999999995 or 24.5 < w

   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. associate-*l/ 99.1%

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

      3. *-lft-identity 99.1%

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

   3. Simplified 99.1%

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

      1. add-sqr-sqrt 99.1%

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

      2. unpow-prod-down 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. pow-sqr 99.1%

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

      2. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. add-exp-log 99.1%

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

      2. log-div 99.1%

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

      3. pow-unpow 99.1%

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

      4. unpow2 99.1%

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

      5. unpow-prod-down 99.1%

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

      6. add-sqr-sqrt 99.1%

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

      7. log-pow 99.1%

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

      8. add-log-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in w around inf 99.1%

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

       1. neg-mul-1 99.1%

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

   12. Simplified 99.1%

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

   if -0.68999999999999995 < w < 24.5

   1. Initial program 99.3%

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

   2. Taylor expanded in w around 0 95.6%

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

   3. Taylor expanded in w around 0 95.6%

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

      1. +-commutative 95.6%

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

      2. mul-1-neg 95.6%

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

      3. unsub-neg 95.6%

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

   5. Simplified 95.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.69 \lor \neg \left(w \leq 24.5\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - w \cdot \ell\\ \end{array} \]

Alternative 4: 97.6% 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 99.2%

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

   1. exp-neg 99.2%

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

   2. associate-*l/ 99.2%

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

   3. *-lft-identity 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in w around 0 96.8%

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

5. Final simplification 96.8%

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

Alternative 5: 64.9% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.45:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -2.45) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.45) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-2.45d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -2.45) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.45:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.45)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -2.45)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.45], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -2.4500000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in w around 0 98.7%

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

   3. Taylor expanded in w around 0 26.1%

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

      1. +-commutative 26.1%

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

      2. mul-1-neg 26.1%

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

      3. unsub-neg 26.1%

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

   5. Simplified 26.1%

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

   6. Taylor expanded in w around inf 26.1%

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

      1. mul-1-neg 26.1%

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

      2. distribute-lft-neg-out 26.1%

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

      3. *-commutative 26.1%

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

   8. Simplified 26.1%

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

   if -2.4500000000000002 < w

   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. associate-*l/ 98.9%

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

      3. *-lft-identity 98.9%

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

   3. Simplified 98.9%

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

      1. add-sqr-sqrt 98.3%

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

      2. unpow-prod-down 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. pow-sqr 98.3%

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

      2. *-commutative 98.3%

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

   7. Simplified 98.3%

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

      1. *-un-lft-identity 98.3%

         \[\leadsto \frac{\color{blue}{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}}{e^{w}} \]

      2. add-cbrt-cube 98.3%

         \[\leadsto \frac{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\color{blue}{\sqrt[3]{\left(e^{w} \cdot e^{w}\right) \cdot e^{w}}}} \]

      3. exp-sum 98.3%

         \[\leadsto \frac{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{\color{blue}{e^{w + w}} \cdot e^{w}}} \]

      4. cbrt-prod 98.3%

         \[\leadsto \frac{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\color{blue}{\sqrt[3]{e^{w + w}} \cdot \sqrt[3]{e^{w}}}} \]

      5. times-frac 98.3%

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{w + w}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}}} \]

      6. exp-sum 98.3%

         \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{e^{w} \cdot e^{w}}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}} \]

      7. cbrt-prod 98.3%

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}} \]

      8. pow2 98.3%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}} \]

      9. pow-unpow 98.3%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\left({\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}\right)}^{2}}}{\sqrt[3]{e^{w}}} \]

      10. unpow2 98.3%

          \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

      11. unpow-prod-down 98.3%

          \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

      12. add-sqr-sqrt 98.9%

          \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\color{blue}{\ell}}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

   9. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

   10. Taylor expanded in w around 0 78.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.45:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 6: 64.6% 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 99.2%

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

2. Taylor expanded in w around 0 96.8%

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

3. Taylor expanded in w around 0 62.4%

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

   1. +-commutative 62.4%

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

   2. mul-1-neg 62.4%

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

   3. unsub-neg 62.4%

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

5. Simplified 62.4%

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

6. Taylor expanded in l around 0 62.4%

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

7. Final simplification 62.4%

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

Alternative 7: 64.6% 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 99.2%

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

2. Taylor expanded in w around 0 96.8%

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

3. Taylor expanded in w around 0 62.4%

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

   1. +-commutative 62.4%

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

   2. mul-1-neg 62.4%

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

   3. unsub-neg 62.4%

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

5. Simplified 62.4%

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

6. Final simplification 62.4%

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

Alternative 8: 57.7% 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 99.2%

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

   1. exp-neg 99.2%

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

   2. associate-*l/ 99.2%

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

   3. *-lft-identity 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 98.8%

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

   2. unpow-prod-down 98.8%

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

5. Applied egg-rr 98.8%

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

   1. pow-sqr 98.8%

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

   2. *-commutative 98.8%

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

7. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

      \[\leadsto \frac{\color{blue}{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}}{e^{w}} \]

   2. add-cbrt-cube 98.4%

      \[\leadsto \frac{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\color{blue}{\sqrt[3]{\left(e^{w} \cdot e^{w}\right) \cdot e^{w}}}} \]

   3. exp-sum 98.4%

      \[\leadsto \frac{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{\color{blue}{e^{w + w}} \cdot e^{w}}} \]

   4. cbrt-prod 98.4%

      \[\leadsto \frac{1 \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\color{blue}{\sqrt[3]{e^{w + w}} \cdot \sqrt[3]{e^{w}}}} \]

   5. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{w + w}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}}} \]

   6. exp-sum 98.4%

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{e^{w} \cdot e^{w}}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}} \]

   7. cbrt-prod 98.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}} \]

   8. pow2 98.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\left(\sqrt{\ell}\right)}^{\left(e^{w} \cdot 2\right)}}{\sqrt[3]{e^{w}}} \]

   9. pow-unpow 98.8%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\left({\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}\right)}^{2}}}{\sqrt[3]{e^{w}}} \]

   10. unpow2 98.8%

       \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

   11. unpow-prod-down 98.8%

       \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

   12. add-sqr-sqrt 99.2%

       \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\color{blue}{\ell}}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

9. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

10. Taylor expanded in w around 0 56.1%

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

11. Final simplification 56.1%

    \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023258 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))