exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 11.6s

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: 99.5% 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 99.3%

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

2. Final simplification 99.3%

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

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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 97.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{-1}{-e^{w}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l * (-1.0 / -math.exp(w))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in w around 0 98.3%

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

   1. frac-2neg 98.3%

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

   2. div-inv 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

   \[\leadsto \ell \cdot \frac{-1}{-e^{w}} \]

Alternative 4: 97.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in w around 0 98.3%

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

5. Final simplification 98.3%

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

Alternative 5: 67.1% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-117}:\\ \;\;\;\;\ell\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(w \cdot \ell\right) \cdot \left(w \cdot \ell\right)}{\ell + w \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 8e-117)
   l
   (if (<= l 3e+151)
     (/ (- (* l l) (* (* w l) (* w l))) (+ l (* w l)))
     (* l (- 1.0 w)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 8e-117) {
		tmp = l;
	} else if (l <= 3e+151) {
		tmp = ((l * l) - ((w * l) * (w * l))) / (l + (w * l));
	} else {
		tmp = l * (1.0 - 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 <= 8d-117) then
        tmp = l
    else if (l <= 3d+151) then
        tmp = ((l * l) - ((w * l) * (w * l))) / (l + (w * l))
    else
        tmp = l * (1.0d0 - w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (l <= 8e-117) {
		tmp = l;
	} else if (l <= 3e+151) {
		tmp = ((l * l) - ((w * l) * (w * l))) / (l + (w * l));
	} else {
		tmp = l * (1.0 - w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if l <= 8e-117:
		tmp = l
	elif l <= 3e+151:
		tmp = ((l * l) - ((w * l) * (w * l))) / (l + (w * l))
	else:
		tmp = l * (1.0 - w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 8e-117)
		tmp = l;
	elseif (l <= 3e+151)
		tmp = Float64(Float64(Float64(l * l) - Float64(Float64(w * l) * Float64(w * l))) / Float64(l + Float64(w * l)));
	else
		tmp = Float64(l * Float64(1.0 - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (l <= 8e-117)
		tmp = l;
	elseif (l <= 3e+151)
		tmp = ((l * l) - ((w * l) * (w * l))) / (l + (w * l));
	else
		tmp = l * (1.0 - w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[l, 8e-117], l, If[LessEqual[l, 3e+151], N[(N[(N[(l * l), $MachinePrecision] - N[(N[(w * l), $MachinePrecision] * N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l + N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.00000000000000024e-117

   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. 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. *-un-lft-identity 99.1%

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

      2. add-cube-cbrt 99.1%

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

      3. times-frac 99.1%

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

      4. pow2 99.1%

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

      5. pow-unpow 99.1%

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

      6. unpow2 99.1%

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

      7. unpow-prod-down 99.1%

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

      8. add-sqr-sqrt 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in w around 0 53.8%

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

   if 8.00000000000000024e-117 < l < 2.9999999999999999e151

   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 98.7%

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

   5. Taylor expanded in w around 0 56.3%

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

      1. +-commutative 56.3%

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

      2. mul-1-neg 56.3%

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

      3. unsub-neg 56.3%

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

   7. Simplified 56.3%

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

      1. sub-neg 56.3%

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

      2. flip-+ 69.1%

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

      3. *-commutative 69.1%

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

      4. distribute-rgt-neg-in 69.1%

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

      5. *-commutative 69.1%

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

      6. distribute-rgt-neg-in 69.1%

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

      7. *-commutative 69.1%

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

      8. distribute-rgt-neg-in 69.1%

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

   9. Applied egg-rr 69.1%

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

   if 2.9999999999999999e151 < l

   1. Initial program 98.3%

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

      1. exp-neg 98.3%

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

      2. associate-*l/ 98.3%

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

      3. *-lft-identity 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in w around 0 97.8%

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

   5. Taylor expanded in w around 0 85.4%

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

      1. +-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. unsub-neg 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in l around 0 85.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-117}:\\ \;\;\;\;\ell\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\frac{\ell \cdot \ell - \left(w \cdot \ell\right) \cdot \left(w \cdot \ell\right)}{\ell + w \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \end{array} \]

Alternative 6: 64.1% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.035:\\ \;\;\;\;\left(-w\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.035) (* (- w) l) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.035) {
		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 <= (-0.035d0)) 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 <= -0.035) {
		tmp = -w * l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.035:
		tmp = -w * l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.035)
		tmp = Float64(Float64(-w) * l);
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.035)
		tmp = -w * l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.035], N[((-w) * l), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.035000000000000003

   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 28.3%

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

      1. +-commutative 28.3%

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

      2. mul-1-neg 28.3%

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

      3. unsub-neg 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in w around inf 28.3%

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

      1. associate-*r* 28.3%

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

      2. neg-mul-1 28.3%

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

      3. *-commutative 28.3%

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

   10. Simplified 28.3%

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

   if -0.035000000000000003 < 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-cube-cbrt 98.3%

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

      3. times-frac 98.3%

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

      4. pow2 98.3%

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

      5. pow-unpow 98.3%

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

      6. unpow2 98.3%

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

      7. unpow-prod-down 98.3%

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

      8. 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 81.8%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.035:\\ \;\;\;\;\left(-w\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 7: 63.8% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in w around 0 98.3%

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

5. Taylor expanded in w around 0 63.6%

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

   1. +-commutative 63.6%

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

   2. mul-1-neg 63.6%

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

   3. unsub-neg 63.6%

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

7. Simplified 63.6%

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

8. Taylor expanded in l around 0 63.6%

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

9. Final simplification 63.6%

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

Alternative 8: 63.8% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in w around 0 98.3%

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

5. Taylor expanded in w around 0 63.6%

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

   1. +-commutative 63.6%

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

   2. mul-1-neg 63.6%

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

   3. unsub-neg 63.6%

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

7. Simplified 63.6%

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

8. Final simplification 63.6%

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

Alternative 9: 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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 98.9%

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

   2. unpow-prod-down 98.9%

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

5. Applied egg-rr 98.9%

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

   1. pow-sqr 98.9%

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

   2. *-commutative 98.9%

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

7. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. add-cube-cbrt 98.9%

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

   3. times-frac 98.9%

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

   4. pow2 98.9%

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

   5. pow-unpow 98.9%

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

   6. unpow2 98.9%

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

   7. unpow-prod-down 98.9%

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

   8. add-sqr-sqrt 99.3%

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

9. Applied egg-rr 99.3%

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

10. Taylor expanded in w around 0 55.7%

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

11. Final simplification 55.7%

    \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023238 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))