exp-w (used to crash)

Percentage Accurate: 99.5% → 99.4%

Time: 13.0s

Alternatives: 7

Speedup: N/A×

• 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left(e^{w \cdot 0.5}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/ (pow (pow l (sqrt (exp w))) (exp (* w 0.5))) (exp w)))

C

double code(double w, double l) {
	return pow(pow(l, sqrt(exp(w))), exp((w * 0.5))) / exp(w);
}

Fortran

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

Java

public static double code(double w, double l) {
	return Math.pow(Math.pow(l, Math.sqrt(Math.exp(w))), Math.exp((w * 0.5))) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(math.pow(l, math.sqrt(math.exp(w))), math.exp((w * 0.5))) / math.exp(w)

Julia

function code(w, l)
	return Float64(((l ^ sqrt(exp(w))) ^ exp(Float64(w * 0.5))) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l ^ sqrt(exp(w))) ^ exp((w * 0.5))) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[N[Power[l, N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Exp[N[(w * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

   1. add-exp-log 94.6%

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

   2. log-pow 94.6%

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

5. Applied egg-rr 94.6%

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

   1. *-commutative 94.6%

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

   2. pow-to-exp 98.5%

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

   3. add-sqr-sqrt 98.4%

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

   4. pow-unpow 98.4%

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

7. Applied egg-rr 98.4%

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

   1. pow1/2 98.4%

      \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left({\left(e^{w}\right)}^{0.5}\right)}}}{e^{w}} \]

   2. pow-exp 98.5%

      \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left(e^{w \cdot 0.5}\right)}}}{e^{w}} \]

9. Applied egg-rr 98.5%

   \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left(e^{w \cdot 0.5}\right)}}}{e^{w}} \]

10. Final simplification 98.5%

    \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left(e^{w \cdot 0.5}\right)}}{e^{w}} \]

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 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 3: 97.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 440000000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 440000000.0)))
   (exp (- w))
   (- l (+ (* l w) (* (* w w) (* l -0.5))))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 440000000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.7d0)) .or. (.not. (w <= 440000000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l - ((l * w) + ((w * w) * (l * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 440000000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 440000000.0):
		tmp = math.exp(-w)
	else:
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 440000000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l - Float64(Float64(l * w) + Float64(Float64(w * w) * Float64(l * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 440000000.0)))
		tmp = exp(-w);
	else
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 440000000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l - N[(N[(l * w), $MachinePrecision] + N[(N[(w * w), $MachinePrecision] * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 4.4e8 < 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-exp-log 100.0%

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

      2. log-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. pow-to-exp 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. pow-unpow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-div 100.0%

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

      3. pow-pow 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. log-pow 100.0%

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

      6. 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 100.0%

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

       1. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

   if -0.69999999999999996 < w < 4.4e8

   1. Initial program 96.9%

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

      1. exp-neg 96.9%

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

      2. associate-*l/ 96.9%

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

      3. *-lft-identity 96.9%

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

   3. Simplified 96.9%

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

      1. add-exp-log 89.2%

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

      2. log-pow 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. *-commutative 89.2%

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

      2. pow-to-exp 96.9%

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

      3. add-sqr-sqrt 96.9%

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

      4. pow-unpow 96.9%

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

   7. Applied egg-rr 96.9%

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

   8. Taylor expanded in w around 0 94.4%

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

   9. Taylor expanded in w around 0 94.5%

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

       1. distribute-lft-out 94.5%

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

       2. unpow2 94.5%

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

       3. distribute-rgt-out 94.5%

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

       4. metadata-eval 94.5%

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

   11. Simplified 94.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 440000000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4: 97.4% 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 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

   1. add-exp-log 94.6%

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

   2. log-pow 94.6%

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

5. Applied egg-rr 94.6%

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

   1. *-commutative 94.6%

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

   2. pow-to-exp 98.5%

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

   3. add-sqr-sqrt 98.4%

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

   4. pow-unpow 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in w around 0 97.2%

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

9. Final simplification 97.2%

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

Alternative 5: 73.8% accurate, 23.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

   1. add-exp-log 94.6%

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

   2. log-pow 94.6%

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

5. Applied egg-rr 94.6%

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

   1. *-commutative 94.6%

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

   2. pow-to-exp 98.5%

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

   3. add-sqr-sqrt 98.4%

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

   4. pow-unpow 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in w around 0 97.2%

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

9. Taylor expanded in w around 0 68.5%

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

    1. distribute-lft-out 68.5%

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

    2. unpow2 68.5%

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

    3. distribute-rgt-out 68.5%

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

    4. metadata-eval 68.5%

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

11. Simplified 68.5%

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

12. Final simplification 68.5%

    \[\leadsto \ell - \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right) \]

Alternative 6: 63.6% 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 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

   1. add-exp-log 94.6%

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

   2. log-pow 94.6%

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

5. Applied egg-rr 94.6%

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

   1. *-commutative 94.6%

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

   2. pow-to-exp 98.5%

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

   3. add-sqr-sqrt 98.4%

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

   4. pow-unpow 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in w around 0 97.2%

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

9. Taylor expanded in w around 0 59.2%

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

    1. mul-1-neg 59.2%

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

    2. unsub-neg 59.2%

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

11. Simplified 59.2%

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

12. Final simplification 59.2%

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

Alternative 7: 57.0% 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 98.5%

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

2. Taylor expanded in w around 0 49.5%

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

3. Final simplification 49.5%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023297 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))