exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 41.0s

Alternatives: 10

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: 99.3% 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: 99.3% 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.3% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;{e}^{\left(-w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(\left(w + 1\right) + {w}^{2} \cdot 0.5\right)}}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0)
   (pow E (- w))
   (/ (pow l (+ (+ w 1.0) (* (pow w 2.0) 0.5))) (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = pow(((double) M_E), -w);
	} else {
		tmp = pow(l, ((w + 1.0) + (pow(w, 2.0) * 0.5))) / (w + 1.0);
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = Math.pow(Math.E, -w);
	} else {
		tmp = Math.pow(l, ((w + 1.0) + (Math.pow(w, 2.0) * 0.5))) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = math.pow(math.e, -w)
	else:
		tmp = math.pow(l, ((w + 1.0) + (math.pow(w, 2.0) * 0.5))) / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(1) ^ Float64(-w);
	else
		tmp = Float64((l ^ Float64(Float64(w + 1.0) + Float64((w ^ 2.0) * 0.5))) / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = 2.71828182845904523536 ^ -w;
	else
		tmp = (l ^ ((w + 1.0) + ((w ^ 2.0) * 0.5))) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Power[E, (-w)], $MachinePrecision], N[(N[Power[l, N[(N[(w + 1.0), $MachinePrecision] + N[(N[Power[w, 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 99.9%

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

      5. exp-1-e 99.9%

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

      6. log-div 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. fma-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in w around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   9. Simplified 99.9%

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

   if -1 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 99.5%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in w around 0 99.5%

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

      1. associate-+r+ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;{e}^{\left(-w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(\left(w + 1\right) + {w}^{2} \cdot 0.5\right)}}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;{e}^{\left(-w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0) (pow E (- w)) (/ (pow l (exp w)) (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = pow(((double) M_E), -w);
	} else {
		tmp = pow(l, exp(w)) / (w + 1.0);
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = Math.pow(Math.E, -w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = math.pow(math.e, -w)
	else:
		tmp = math.pow(l, math.exp(w)) / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(1) ^ Float64(-w);
	else
		tmp = Float64((l ^ exp(w)) / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = 2.71828182845904523536 ^ -w;
	else
		tmp = (l ^ exp(w)) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Power[E, (-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 99.9%

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

      5. exp-1-e 99.9%

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

      6. log-div 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. fma-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in w around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   9. Simplified 99.9%

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

   if -1 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 99.5%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.98:\\ \;\;\;\;{e}^{\left(-w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(w + 1\right)}}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.98) (pow E (- w)) (/ (pow l (+ w 1.0)) (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.98) {
		tmp = pow(((double) M_E), -w);
	} else {
		tmp = pow(l, (w + 1.0)) / (w + 1.0);
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.98) {
		tmp = Math.pow(Math.E, -w);
	} else {
		tmp = Math.pow(l, (w + 1.0)) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.98:
		tmp = math.pow(math.e, -w)
	else:
		tmp = math.pow(l, (w + 1.0)) / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.98)
		tmp = exp(1) ^ Float64(-w);
	else
		tmp = Float64((l ^ Float64(w + 1.0)) / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.98)
		tmp = 2.71828182845904523536 ^ -w;
	else
		tmp = (l ^ (w + 1.0)) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.98], N[Power[E, (-w)], $MachinePrecision], N[(N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision] / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.97999999999999998

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 99.9%

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

      5. exp-1-e 99.9%

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

      6. log-div 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. fma-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in w around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   9. Simplified 99.9%

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

   if -0.97999999999999998 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 99.5%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in w around 0 99.4%

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

      1. +-commutative 99.0%

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

   10. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.98:\\ \;\;\;\;{e}^{\left(-w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(w + 1\right)}}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0) (pow E (- w)) (* l (pow l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = pow(((double) M_E), -w);
	} else {
		tmp = l * pow(l, w);
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = Math.pow(Math.E, -w);
	} else {
		tmp = l * Math.pow(l, w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = math.pow(math.e, -w)
	else:
		tmp = l * math.pow(l, w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(1) ^ Float64(-w);
	else
		tmp = Float64(l * (l ^ w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = 2.71828182845904523536 ^ -w;
	else
		tmp = l * (l ^ w);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 99.9%

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

      5. exp-1-e 99.9%

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

      6. log-div 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. fma-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in w around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   9. Simplified 99.9%

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

   if -1 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 99.0%

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

   6. Taylor expanded in w around 0 99.0%

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

      1. +-commutative 99.0%

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

   8. Simplified 99.0%

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

      1. pow-plus 99.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{w} \cdot \ell}}{1} \]

   10. Applied egg-rr 99.0%

       \[\leadsto \frac{\color{blue}{{\ell}^{w} \cdot \ell}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternative 7: 97.5% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

6. Final simplification 97.6%

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

Alternative 8: 76.9% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.116:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.116) (- l (* w l)) (+ -1.0 (+ l 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.116) {
		tmp = l - (w * l);
	} else {
		tmp = -1.0 + (l + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.116d0) then
        tmp = l - (w * l)
    else
        tmp = (-1.0d0) + (l + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.116) {
		tmp = l - (w * l);
	} else {
		tmp = -1.0 + (l + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.116:
		tmp = l - (w * l)
	else:
		tmp = -1.0 + (l + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.116)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(-1.0 + Float64(l + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.116)
		tmp = l - (w * l);
	else
		tmp = -1.0 + (l + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.116], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(l + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.116000000000000006

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 98.5%

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

   6. Taylor expanded in w around 0 72.8%

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

      1. mul-1-neg 72.8%

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

      2. unsub-neg 72.8%

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

      3. *-commutative 72.8%

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

   8. Simplified 72.8%

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

   if 0.116000000000000006 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

      2. expm1-udef 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{e^{\mathsf{log1p}\left(w\right)} - 1}}} \]

      3. exp-diff 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\frac{e^{e^{\mathsf{log1p}\left(w\right)}}}{e^{1}}}} \]

      4. log1p-udef 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-1-e 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0 5.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot e}{e^{1}}} \]
   8. Step-by-step derivation

      1. exp-1-e 5.2%

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

      2. associate-/l* 5.2%

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

      3. *-inverses 5.2%

         \[\leadsto \frac{\ell}{\color{blue}{1}} \]

      4. rem-square-sqrt 5.2%

         \[\leadsto \frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{1} \]

      5. associate-*r/ 5.2%

         \[\leadsto \color{blue}{\sqrt{\ell} \cdot \frac{\sqrt{\ell}}{1}} \]

      6. /-rgt-identity 5.2%

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

      7. rem-square-sqrt 5.2%

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

   9. Simplified 5.2%

      \[\leadsto \color{blue}{\ell} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 5.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell\right)\right)} \]

       2. expm1-udef 91.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]

       3. log1p-udef 91.2%

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

       4. rem-exp-log 91.2%

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

   11. Applied egg-rr 91.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.116:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.6%

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

6. Taylor expanded in w around 0 63.9%

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

   1. mul-1-neg 63.9%

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

   2. unsub-neg 63.9%

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

   3. *-commutative 63.9%

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

8. Simplified 63.9%

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

9. Final simplification 63.9%

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

Alternative 10: 57.5% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-lft-identity 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 68.2%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w\right)\right)}}} \]

   2. expm1-udef 68.2%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{e^{\mathsf{log1p}\left(w\right)} - 1}}} \]

   3. exp-diff 68.2%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\frac{e^{e^{\mathsf{log1p}\left(w\right)}}}{e^{1}}}} \]

   4. log1p-udef 68.2%

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

   5. rem-exp-log 99.8%

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

   6. exp-1-e 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in w around 0 56.3%

   \[\leadsto \color{blue}{\frac{\ell \cdot e}{e^{1}}} \]
8. Step-by-step derivation

   1. exp-1-e 56.3%

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

   2. associate-/l* 56.5%

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

   3. *-inverses 56.5%

      \[\leadsto \frac{\ell}{\color{blue}{1}} \]

   4. rem-square-sqrt 56.0%

      \[\leadsto \frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{1} \]

   5. associate-*r/ 56.0%

      \[\leadsto \color{blue}{\sqrt{\ell} \cdot \frac{\sqrt{\ell}}{1}} \]

   6. /-rgt-identity 56.0%

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

   7. rem-square-sqrt 56.5%

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

9. Simplified 56.5%

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

10. Final simplification 56.5%

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

Reproduce

herbie shell --seed 2024101 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))