exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 12.7s

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 99.0%

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

2. Final simplification 99.0%

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

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 99.0%

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

   1. exp-neg 99.0%

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

   2. associate-*l/ 99.0%

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

   3. *-lft-identity 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 3: 97.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 14500000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 14500000.0))) (exp (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 14500000.0)) {
		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 <= (-0.7d0)) .or. (.not. (w <= 14500000.0d0))) 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 <= -0.7) || !(w <= 14500000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 14500000.0):
		tmp = math.exp(-w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 14500000.0))
		tmp = exp(Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 14500000.0)))
		tmp = exp(-w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 14500000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 1.45e7 < 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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. log-rec 100.0%

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

      5. remove-double-div 100.0%

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

   6. Simplified 100.0%

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

      1. div-exp 100.0%

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

      2. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in w around inf 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -0.69999999999999996 < w < 1.45e7

   1. Initial program 98.2%

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

   2. Taylor expanded in w around 0 97.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 14500000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

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 99.0%

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

   1. exp-neg 99.0%

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

   2. associate-*l/ 99.0%

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

   3. *-lft-identity 99.0%

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

3. Simplified 99.0%

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

   1. add-sqr-sqrt 47.5%

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

   2. sqrt-unprod 84.1%

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

   3. sqr-neg 84.1%

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

   4. sqrt-unprod 36.6%

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

   5. add-sqr-sqrt 83.9%

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

   6. add-sqr-sqrt 83.9%

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

   7. sqrt-unprod 83.9%

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

   8. add-sqr-sqrt 36.6%

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

   9. sqrt-unprod 66.7%

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

   10. sqr-neg 66.7%

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

   11. sqrt-unprod 30.1%

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

   12. add-sqr-sqrt 55.0%

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

   13. pow1 55.0%

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

   14. exp-neg 55.0%

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

   15. inv-pow 55.0%

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

   16. pow-prod-up 97.6%

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

   17. metadata-eval 97.6%

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

   18. metadata-eval 97.6%

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

   19. metadata-eval 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in l around 0 97.6%

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

7. Final simplification 97.6%

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

Alternative 5: 77.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.048:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + 1\right) \cdot \left(\ell + 1\right) + -1}{1 + \left(\ell + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.048)
   (* l (- 1.0 w))
   (/ (+ (* (+ l 1.0) (+ l 1.0)) -1.0) (+ 1.0 (+ l 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.048) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (((l + 1.0) * (l + 1.0)) + -1.0) / (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 <= 0.048d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = (((l + 1.0d0) * (l + 1.0d0)) + (-1.0d0)) / (1.0d0 + (l + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.048) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (((l + 1.0) * (l + 1.0)) + -1.0) / (1.0 + (l + 1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.048:
		tmp = l * (1.0 - w)
	else:
		tmp = (((l + 1.0) * (l + 1.0)) + -1.0) / (1.0 + (l + 1.0))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.048)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(Float64(Float64(Float64(l + 1.0) * Float64(l + 1.0)) + -1.0) / Float64(1.0 + Float64(l + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.048)
		tmp = l * (1.0 - w);
	else
		tmp = (((l + 1.0) * (l + 1.0)) + -1.0) / (1.0 + (l + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.048], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + 1.0), $MachinePrecision] * N[(l + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(l + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.048000000000000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 37.0%

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

      2. sqrt-unprod 81.6%

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

      3. sqr-neg 81.6%

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

      4. sqrt-unprod 44.6%

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

      5. add-sqr-sqrt 81.3%

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

      6. add-sqr-sqrt 81.3%

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

      7. sqrt-unprod 81.3%

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

      8. add-sqr-sqrt 44.6%

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

      9. sqrt-unprod 81.3%

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

      10. sqr-neg 81.3%

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

      11. sqrt-unprod 36.7%

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

      12. add-sqr-sqrt 67.0%

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

      13. pow1 67.0%

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

      14. exp-neg 67.0%

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

      15. inv-pow 67.0%

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

      16. pow-prod-up 98.0%

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

      17. metadata-eval 98.0%

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

      18. metadata-eval 98.0%

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

      19. metadata-eval 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in l around 0 98.0%

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

   7. Taylor expanded in w around 0 74.7%

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

      1. mul-1-neg 74.7%

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

      2. *-lft-identity 74.7%

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

      3. distribute-rgt-neg-out 74.7%

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

      4. *-commutative 74.7%

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

      5. distribute-rgt-in 74.7%

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

      6. sub-neg 74.7%

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

   9. Simplified 74.7%

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

   if 0.048000000000000001 < w

   1. Initial program 95.7%

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

      1. add-sqr-sqrt 95.7%

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

      2. sqrt-unprod 95.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. pow1 0.0%

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

      9. exp-neg 0.0%

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

      10. inv-pow 0.0%

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

      11. pow-prod-up 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-un-lft-identity 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. sqrt-unprod 100.0%

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

      18. sqr-neg 100.0%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 3.1%

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

   3. Applied egg-rr 5.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 91.7%

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

      2. flip-- 93.7%

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

      3. log1p-udef 93.7%

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

      4. rem-exp-log 93.7%

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

      5. log1p-udef 93.7%

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

      6. rem-exp-log 93.7%

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

      7. +-commutative 93.7%

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

      8. +-commutative 93.7%

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

      9. metadata-eval 93.7%

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

      10. log1p-udef 93.7%

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

      11. rem-exp-log 93.7%

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

      12. +-commutative 93.7%

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

   5. Applied egg-rr 93.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.048:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + 1\right) \cdot \left(\ell + 1\right) + -1}{1 + \left(\ell + 1\right)}\\ \end{array} \]

Alternative 6: 77.1% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.094:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.094) (* l (- 1.0 w)) (+ (+ l 1.0) -1.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.094) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (l + 1.0) + -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 <= 0.094d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = (l + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.094) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (l + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.094:
		tmp = l * (1.0 - w)
	else:
		tmp = (l + 1.0) + -1.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.094)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(Float64(l + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.094)
		tmp = l * (1.0 - w);
	else
		tmp = (l + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.094], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(N[(l + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.094

   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. Step-by-step derivation

      1. add-sqr-sqrt 37.0%

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

      2. sqrt-unprod 81.6%

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

      3. sqr-neg 81.6%

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

      4. sqrt-unprod 44.6%

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

      5. add-sqr-sqrt 81.3%

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

      6. add-sqr-sqrt 81.3%

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

      7. sqrt-unprod 81.3%

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

      8. add-sqr-sqrt 44.6%

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

      9. sqrt-unprod 81.3%

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

      10. sqr-neg 81.3%

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

      11. sqrt-unprod 36.7%

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

      12. add-sqr-sqrt 67.0%

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

      13. pow1 67.0%

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

      14. exp-neg 67.0%

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

      15. inv-pow 67.0%

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

      16. pow-prod-up 98.0%

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

      17. metadata-eval 98.0%

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

      18. metadata-eval 98.0%

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

      19. metadata-eval 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in l around 0 98.0%

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

   7. Taylor expanded in w around 0 74.7%

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

      1. mul-1-neg 74.7%

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

      2. *-lft-identity 74.7%

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

      3. distribute-rgt-neg-out 74.7%

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

      4. *-commutative 74.7%

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

      5. distribute-rgt-in 74.7%

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

      6. sub-neg 74.7%

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

   9. Simplified 74.7%

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

   if 0.094 < w

   1. Initial program 95.7%

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

      1. exp-neg 95.7%

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

      2. associate-/r/ 95.7%

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

      3. clear-num 95.7%

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

      4. clear-num 95.7%

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

      5. associate-/r/ 95.7%

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

      6. exp-neg 95.7%

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

      7. add-sqr-sqrt 95.7%

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

      8. sqrt-unprod 95.7%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. sqr-neg 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. pow1 0.0%

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

      15. exp-neg 0.0%

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

      16. inv-pow 0.0%

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

      17. pow-prod-up 100.0%

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

      18. metadata-eval 100.0%

          \[\leadsto \frac{1}{\frac{1}{\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}} \cdot {\ell}^{\left(e^{w}\right)}}} \]

      19. metadata-eval 100.0%

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

      20. metadata-eval 100.0%

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

   3. Applied egg-rr 5.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\ell}}} \]
   4. Step-by-step derivation

      1. remove-double-div 5.2%

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

      2. expm1-log1p-u 5.2%

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

      3. expm1-udef 91.7%

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

      4. log1p-udef 91.7%

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

      5. rem-exp-log 91.7%

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

      6. +-commutative 91.7%

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

   5. Applied egg-rr 91.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.094:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \]

Alternative 7: 65.0% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -160:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -160.0) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -160.0) {
		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 <= (-160.0d0)) 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 <= -160.0) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -160.0:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -160.0)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -160.0)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -160.0], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -160

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 45.5%

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

      3. sqr-neg 45.5%

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

      4. sqrt-unprod 45.5%

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

      5. add-sqr-sqrt 45.5%

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

      6. add-sqr-sqrt 45.5%

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

      7. sqrt-unprod 45.5%

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

      8. add-sqr-sqrt 45.5%

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

      9. sqrt-unprod 45.5%

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

      10. sqr-neg 45.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. pow1 0.1%

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

      14. exp-neg 0.1%

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

      15. inv-pow 0.1%

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

      16. pow-prod-up 98.5%

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

      17. metadata-eval 98.5%

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

      18. metadata-eval 98.5%

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

      19. metadata-eval 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in w around 0 24.4%

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

      1. mul-1-neg 24.4%

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

      2. unsub-neg 24.4%

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

   8. Simplified 24.4%

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

   9. Taylor expanded in w around inf 24.4%

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

       1. mul-1-neg 24.4%

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

       2. *-commutative 24.4%

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

       3. distribute-rgt-neg-in 24.4%

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

   11. Simplified 24.4%

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

   if -160 < w

   1. Initial program 98.6%

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

   2. Taylor expanded in w around 0 75.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -160:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 8: 64.7% 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 99.0%

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

   1. exp-neg 99.0%

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

   2. associate-*l/ 99.0%

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

   3. *-lft-identity 99.0%

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

3. Simplified 99.0%

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

   1. add-sqr-sqrt 47.5%

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

   2. sqrt-unprod 84.1%

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

   3. sqr-neg 84.1%

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

   4. sqrt-unprod 36.6%

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

   5. add-sqr-sqrt 83.9%

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

   6. add-sqr-sqrt 83.9%

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

   7. sqrt-unprod 83.9%

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

   8. add-sqr-sqrt 36.6%

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

   9. sqrt-unprod 66.7%

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

   10. sqr-neg 66.7%

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

   11. sqrt-unprod 30.1%

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

   12. add-sqr-sqrt 55.0%

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

   13. pow1 55.0%

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

   14. exp-neg 55.0%

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

   15. inv-pow 55.0%

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

   16. pow-prod-up 97.6%

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

   17. metadata-eval 97.6%

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

   18. metadata-eval 97.6%

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

   19. metadata-eval 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in l around 0 97.6%

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

7. Taylor expanded in w around 0 61.9%

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

   1. mul-1-neg 61.9%

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

   2. *-lft-identity 61.9%

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

   3. distribute-rgt-neg-out 61.9%

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

   4. *-commutative 61.9%

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

   5. distribute-rgt-in 61.9%

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

   6. sub-neg 61.9%

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

9. Simplified 61.9%

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

10. Final simplification 61.9%

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

Alternative 9: 58.1% 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 99.0%

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

2. Taylor expanded in w around 0 56.8%

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

3. Final simplification 56.8%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023309 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))