exp-w (used to crash)

Percentage Accurate: 99.3% → 97.6%

Time: 31.8s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. w = -2.00000000000000007e154
  2. l = 0.0

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: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.108:\\ \;\;\;\;e^{-w} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.108) (* (exp (- w)) l) (log (exp l))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.108) {
		tmp = exp(-w) * l;
	} else {
		tmp = log(exp(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 <= 0.108d0) then
        tmp = exp(-w) * l
    else
        tmp = log(exp(l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.108) {
		tmp = Math.exp(-w) * l;
	} else {
		tmp = Math.log(Math.exp(l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.108:
		tmp = math.exp(-w) * l
	else:
		tmp = math.log(math.exp(l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.108)
		tmp = Float64(exp(Float64(-w)) * l);
	else
		tmp = log(exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.108)
		tmp = exp(-w) * l;
	else
		tmp = log(exp(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.108], N[(N[Exp[(-w)], $MachinePrecision] * l), $MachinePrecision], N[Log[N[Exp[l], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.107999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0 98.3%

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

   if 0.107999999999999999 < w

   1. Initial program 91.3%

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

      1. add-sqr-sqrt 91.3%

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

      2. sqrt-unprod 91.3%

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

      3. exp-neg 91.3%

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

      4. inv-pow 91.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.5%

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

      7. sqr-neg 6.5%

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

      8. sqrt-unprod 6.5%

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

      9. add-sqr-sqrt 6.5%

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

      10. pow1 6.5%

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

      11. pow-prod-up 95.7%

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

      12. metadata-eval 95.7%

          \[\leadsto \sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      13. metadata-eval 95.7%

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

      14. metadata-eval 95.7%

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

      15. *-un-lft-identity 95.7%

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

      16. add-sqr-sqrt 95.7%

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

      17. sqrt-unprod 95.7%

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

      18. sqr-neg 95.7%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 2.9%

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

      21. add-sqr-sqrt 2.9%

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

   4. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{{\left({\ell}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 70.7%

         \[\leadsto \color{blue}{\sqrt[3]{{\ell}^{3}}} \]

      2. rem-cbrt-cube 5.5%

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

      3. add-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.108:\\ \;\;\;\;e^{-w} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \left(\sqrt[3]{e^{-w}} \cdot {\ell}^{\left(e^{w}\right)}\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (* (/ 1.0 (pow (cbrt (exp w)) 2.0)) (* (cbrt (exp (- w))) (pow l (exp w)))))

C

double code(double w, double l) {
	return (1.0 / pow(cbrt(exp(w)), 2.0)) * (cbrt(exp(-w)) * pow(l, exp(w)));
}

Java

public static double code(double w, double l) {
	return (1.0 / Math.pow(Math.cbrt(Math.exp(w)), 2.0)) * (Math.cbrt(Math.exp(-w)) * Math.pow(l, Math.exp(w)));
}

Julia

function code(w, l)
	return Float64(Float64(1.0 / (cbrt(exp(w)) ^ 2.0)) * Float64(cbrt(exp(Float64(-w))) * (l ^ exp(w))))
end

Wolfram

code[w_, l_] := N[(N[(1.0 / N[Power[N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[(-w)], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.2%

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

   1. exp-neg 98.2%

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

   2. remove-double-neg 98.2%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. remove-double-neg 98.2%

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

3. Simplified 98.2%

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

5. Taylor expanded in l around inf 92.8%

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

   1. mul-1-neg 92.8%

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

   2. distribute-rgt-neg-in 92.8%

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

   3. log-rec 92.8%

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

   4. remove-double-div 92.8%

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

7. Simplified 92.8%

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

   1. *-un-lft-identity 92.8%

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

   2. metadata-eval 92.8%

      \[\leadsto \frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot e^{e^{w} \cdot \log \ell}}{e^{w}} \]

   3. add-cube-cbrt 92.8%

      \[\leadsto \frac{\left(3 \cdot 0.3333333333333333\right) \cdot e^{e^{w} \cdot \log \ell}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   4. times-frac 92.8%

      \[\leadsto \color{blue}{\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}}} \]

   5. metadata-eval 92.8%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   6. pow2 92.8%

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

   7. *-commutative 92.8%

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

   8. exp-to-pow 98.2%

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

9. Applied egg-rr 98.2%

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

10. Taylor expanded in l around 0 98.2%

    \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)}\right)} \]
11. Step-by-step derivation

    1. rec-exp 98.2%

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

12. Simplified 98.2%

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

13. Final simplification 98.2%

    \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \left(\sqrt[3]{e^{-w}} \cdot {\ell}^{\left(e^{w}\right)}\right) \]
14. Add Preprocessing

Alternative 3: 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 98.2%

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

3. Final simplification 98.2%

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

Alternative 4: 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 98.2%

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

   1. exp-neg 98.2%

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

   2. remove-double-neg 98.2%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. remove-double-neg 98.2%

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

3. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 31350\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;e^{w} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 31350.0))) (exp (- w)) (* (exp w) l)))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 31350.0)) {
		tmp = exp(-w);
	} else {
		tmp = exp(w) * l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.68d0)) .or. (.not. (w <= 31350.0d0))) then
        tmp = exp(-w)
    else
        tmp = exp(w) * l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 31350.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.exp(w) * l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 31350.0):
		tmp = math.exp(-w)
	else:
		tmp = math.exp(w) * l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 31350.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(exp(w) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 31350.0)))
		tmp = exp(-w);
	else
		tmp = exp(w) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 31350.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(N[Exp[w], $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 31350 < 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. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in l around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. log-rec 99.9%

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

      4. remove-double-div 99.9%

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

   7. Simplified 99.9%

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

      1. div-exp 99.9%

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

      2. fma-neg 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in w around inf 99.2%

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

       1. mul-1-neg 99.2%

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

   12. Simplified 99.2%

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

   if -0.680000000000000049 < w < 31350

   1. Initial program 96.7%

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

   3. Taylor expanded in w around 0 94.2%

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

      1. pow1 94.2%

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

      2. *-commutative 94.2%

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

      3. add-sqr-sqrt 46.9%

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

      4. sqrt-unprod 94.8%

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

      5. sqr-neg 94.8%

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

      6. sqrt-unprod 47.9%

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

      7. add-sqr-sqrt 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. unpow1 94.8%

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

   7. Simplified 94.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 31350\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;e^{w} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 31350\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 31350.0))) (exp (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 31350.0)) {
		tmp = exp(-w);
	} 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 <= (-0.68d0)) .or. (.not. (w <= 31350.0d0))) then
        tmp = exp(-w)
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 31350.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 31350.0):
		tmp = math.exp(-w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 31350.0))
		tmp = exp(Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 31350.0)))
		tmp = exp(-w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 31350.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 31350 < 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. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in l around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. log-rec 99.9%

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

      4. remove-double-div 99.9%

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

   7. Simplified 99.9%

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

      1. div-exp 99.9%

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

      2. fma-neg 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in w around inf 99.2%

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

       1. mul-1-neg 99.2%

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

   12. Simplified 99.2%

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

   if -0.680000000000000049 < w < 31350

   1. Initial program 96.7%

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

   3. Taylor expanded in w around 0 94.2%

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

   4. Taylor expanded in w around 0 93.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 31350\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in w around 0 96.4%

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

4. Final simplification 96.4%

   \[\leadsto e^{-w} \cdot \ell \]
5. Add Preprocessing

Alternative 8: 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 98.2%

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

   1. exp-neg 98.2%

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

   2. remove-double-neg 98.2%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. remove-double-neg 98.2%

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

3. Simplified 98.2%

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

5. Taylor expanded in w around 0 96.4%

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

6. Final simplification 96.4%

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

Alternative 9: 74.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \ell\right) + \ell \cdot 0.5\right) - \ell\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (+ l (* w (- (* w (+ (* -0.16666666666666666 (* w l)) (* l 0.5))) l))))

C

double code(double w, double l) {
	return l + (w * ((w * ((-0.16666666666666666 * (w * l)) + (l * 0.5))) - l));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * ((w * (((-0.16666666666666666d0) * (w * l)) + (l * 0.5d0))) - l))
end function

Java

public static double code(double w, double l) {
	return l + (w * ((w * ((-0.16666666666666666 * (w * l)) + (l * 0.5))) - l));
}

Python

def code(w, l):
	return l + (w * ((w * ((-0.16666666666666666 * (w * l)) + (l * 0.5))) - l))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(Float64(w * Float64(Float64(-0.16666666666666666 * Float64(w * l)) + Float64(l * 0.5))) - l)))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * ((w * ((-0.16666666666666666 * (w * l)) + (l * 0.5))) - l));
end

Wolfram

code[w_, l_] := N[(l + N[(w * N[(N[(w * N[(N[(-0.16666666666666666 * N[(w * l), $MachinePrecision]), $MachinePrecision] + N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in w around 0 96.4%

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

4. Taylor expanded in w around 0 73.7%

   \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \ell + w \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot w\right) + 0.5 \cdot \ell\right)\right)} \]

5. Final simplification 73.7%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \ell\right) + \ell \cdot 0.5\right) - \ell\right) \]
6. Add Preprocessing

Alternative 10: 70.8% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in w around 0 96.4%

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

   1. pow1 96.4%

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

   2. *-commutative 96.4%

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

   3. add-sqr-sqrt 54.9%

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

   4. sqrt-unprod 81.8%

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

   5. sqr-neg 81.8%

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

   6. sqrt-unprod 26.8%

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

   7. add-sqr-sqrt 53.3%

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

5. Applied egg-rr 53.3%

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

   1. unpow1 53.3%

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

7. Simplified 53.3%

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

8. Taylor expanded in w around 0 70.1%

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

   1. associate-*r* 70.1%

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

   2. *-commutative 70.1%

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

10. Simplified 70.1%

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

11. Final simplification 70.1%

    \[\leadsto \ell + w \cdot \left(\ell + w \cdot \left(\ell \cdot 0.5\right)\right) \]
12. Add Preprocessing

Alternative 11: 70.8% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. exp-neg 98.2%

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

   2. remove-double-neg 98.2%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. remove-double-neg 98.2%

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

3. Simplified 98.2%

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

5. Taylor expanded in w around 0 96.4%

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

6. Taylor expanded in w around 0 70.1%

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

   1. associate-*r* 70.1%

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

   2. neg-mul-1 70.1%

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

   3. distribute-rgt-out 70.1%

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

   4. metadata-eval 70.1%

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

8. Simplified 70.1%

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

9. Final simplification 70.1%

   \[\leadsto \ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right) \]
10. Add Preprocessing

Alternative 12: 64.3% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.012:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.012) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.012) {
		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 <= (-0.012d0)) 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 <= -0.012) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.012:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.012)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.012)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.012], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.012

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 98.7%

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

   6. Taylor expanded in w around 0 28.0%

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

      1. mul-1-neg 28.0%

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

      2. unsub-neg 28.0%

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

      3. *-rgt-identity 28.0%

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

      4. distribute-lft-out-- 28.0%

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

   8. Simplified 28.0%

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

   9. Taylor expanded in w around inf 28.0%

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

       1. associate-*r* 28.0%

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

       2. mul-1-neg 28.0%

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

   11. Simplified 28.0%

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

   if -0.012 < w

   1. Initial program 97.4%

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

   3. Taylor expanded in w around 0 95.4%

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

   4. Taylor expanded in w around 0 74.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.012:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% 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 98.2%

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

   1. exp-neg 98.2%

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

   2. remove-double-neg 98.2%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. remove-double-neg 98.2%

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

3. Simplified 98.2%

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

5. Taylor expanded in w around 0 96.4%

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

6. Taylor expanded in w around 0 60.6%

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

   1. mul-1-neg 60.6%

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

   2. unsub-neg 60.6%

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

   3. *-rgt-identity 60.6%

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

   4. distribute-lft-out-- 60.6%

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

8. Simplified 60.6%

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

9. Final simplification 60.6%

   \[\leadsto \ell \cdot \left(1 - w\right) \]
10. Add Preprocessing

Alternative 14: 57.4% 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 98.2%

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

3. Taylor expanded in w around 0 96.4%

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

4. Taylor expanded in w around 0 53.9%

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

5. Final simplification 53.9%

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

Reproduce

herbie shell --seed 2024059 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))