exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 32.2s

Alternatives: 11

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.4% 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 99.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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. Final simplification 99.9%

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

Alternative 3: 97.6% 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 99.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.3%

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

6. Final simplification 98.3%

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

Alternative 4: 88.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.06:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) + w \cdot \left(\left(\ell - \ell \cdot 0.5\right) + \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(\ell + 1\right) \cdot \left(\ell + 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.06)
   (-
    l
    (*
     w
     (+
      l
      (*
       w
       (+
        (- (* l 0.5) l)
        (* w (+ (- l (* l 0.5)) (+ (* l -0.5) (* l 0.16666666666666666)))))))))
   (/ (+ -1.0 (* (+ l 1.0) (+ l 1.0))) 2.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.06) {
		tmp = l - (w * (l + (w * (((l * 0.5) - l) + (w * ((l - (l * 0.5)) + ((l * -0.5) + (l * 0.16666666666666666))))))));
	} else {
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.06d0) then
        tmp = l - (w * (l + (w * (((l * 0.5d0) - l) + (w * ((l - (l * 0.5d0)) + ((l * (-0.5d0)) + (l * 0.16666666666666666d0))))))))
    else
        tmp = ((-1.0d0) + ((l + 1.0d0) * (l + 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.06) {
		tmp = l - (w * (l + (w * (((l * 0.5) - l) + (w * ((l - (l * 0.5)) + ((l * -0.5) + (l * 0.16666666666666666))))))));
	} else {
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.06:
		tmp = l - (w * (l + (w * (((l * 0.5) - l) + (w * ((l - (l * 0.5)) + ((l * -0.5) + (l * 0.16666666666666666))))))))
	else:
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.06)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(Float64(Float64(l * 0.5) - l) + Float64(w * Float64(Float64(l - Float64(l * 0.5)) + Float64(Float64(l * -0.5) + Float64(l * 0.16666666666666666)))))))));
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(l + 1.0) * Float64(l + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.06)
		tmp = l - (w * (l + (w * (((l * 0.5) - l) + (w * ((l - (l * 0.5)) + ((l * -0.5) + (l * 0.16666666666666666))))))));
	else
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.06], N[(l - N[(w * N[(l + N[(w * N[(N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision] + N[(w * N[(N[(l - N[(l * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(l * -0.5), $MachinePrecision] + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(N[(l + 1.0), $MachinePrecision] * N[(l + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.059999999999999998

   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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   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 84.7%

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

   if 0.059999999999999998 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

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

      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 3.8%

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

      7. sqr-neg 3.8%

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

      8. sqrt-unprod 3.8%

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

      9. add-sqr-sqrt 3.8%

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

      10. pow1 3.8%

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

      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%

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

      21. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 62.9%

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

      1. unpow1/3 62.9%

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

      2. rem-cbrt-cube 5.0%

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

      3. expm1-log1p-u 5.0%

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

      4. expm1-undefine 96.4%

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

   6. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 5.0%

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

   8. Simplified 5.0%

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

      1. expm1-undefine 96.4%

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

      2. flip-- 100.0%

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

      3. log1p-undefine 100.0%

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

      4. rem-exp-log 100.0%

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

      5. log1p-undefine 100.0%

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

      6. rem-exp-log 100.0%

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

      7. metadata-eval 100.0%

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

      8. log1p-undefine 100.0%

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

      9. rem-exp-log 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in l around 0 100.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.06:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) + w \cdot \left(\left(\ell - \ell \cdot 0.5\right) + \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(\ell + 1\right) \cdot \left(\ell + 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.075:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.075) (+ l (* l (* w (+ -1.0 (* w 0.5))))) (+ -1.0 (+ l 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.075) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} 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 <= 0.075d0) then
        tmp = l + (l * (w * ((-1.0d0) + (w * 0.5d0))))
    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 <= 0.075) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = -1.0 + (l + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.075:
		tmp = l + (l * (w * (-1.0 + (w * 0.5))))
	else:
		tmp = -1.0 + (l + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.075)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	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 <= 0.075)
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	else
		tmp = -1.0 + (l + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.075], N[(l + N[(l * N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(l + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.0749999999999999972

   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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   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 81.1%

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

      1. associate-*r* 81.1%

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

      2. neg-mul-1 81.1%

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

      3. distribute-rgt-out 81.1%

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

      4. metadata-eval 81.1%

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

   8. Simplified 81.1%

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

   9. Taylor expanded in l around 0 84.7%

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

   if 0.0749999999999999972 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

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

      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 3.8%

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

      7. sqr-neg 3.8%

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

      8. sqrt-unprod 3.8%

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

      9. add-sqr-sqrt 3.8%

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

      10. pow1 3.8%

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

      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%

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

      21. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 62.9%

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

      1. unpow1/3 62.9%

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

      2. rem-cbrt-cube 5.0%

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

      3. expm1-log1p-u 5.0%

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

      4. expm1-undefine 96.4%

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

   6. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 5.0%

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

   8. Simplified 5.0%

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

      1. expm1-undefine 96.4%

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

      2. log1p-undefine 96.4%

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

      3. rem-exp-log 96.4%

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

   10. Applied egg-rr 96.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.075:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.052:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(\ell + 1\right) \cdot \left(\ell + 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.052)
   (+ l (* l (* w (+ -1.0 (* w 0.5)))))
   (/ (+ -1.0 (* (+ l 1.0) (+ l 1.0))) 2.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.052) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.052d0) then
        tmp = l + (l * (w * ((-1.0d0) + (w * 0.5d0))))
    else
        tmp = ((-1.0d0) + ((l + 1.0d0) * (l + 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.052) {
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	} else {
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.052:
		tmp = l + (l * (w * (-1.0 + (w * 0.5))))
	else:
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.052)
		tmp = Float64(l + Float64(l * Float64(w * Float64(-1.0 + Float64(w * 0.5)))));
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(l + 1.0) * Float64(l + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.052)
		tmp = l + (l * (w * (-1.0 + (w * 0.5))));
	else
		tmp = (-1.0 + ((l + 1.0) * (l + 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.052], N[(l + N[(l * N[(w * N[(-1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(N[(l + 1.0), $MachinePrecision] * N[(l + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.0519999999999999976

   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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   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 81.1%

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

      1. associate-*r* 81.1%

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

      2. neg-mul-1 81.1%

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

      3. distribute-rgt-out 81.1%

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

      4. metadata-eval 81.1%

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

   8. Simplified 81.1%

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

   9. Taylor expanded in l around 0 84.7%

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

   if 0.0519999999999999976 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

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

      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 3.8%

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

      7. sqr-neg 3.8%

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

      8. sqrt-unprod 3.8%

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

      9. add-sqr-sqrt 3.8%

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

      10. pow1 3.8%

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

      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%

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

      21. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 62.9%

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

      1. unpow1/3 62.9%

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

      2. rem-cbrt-cube 5.0%

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

      3. expm1-log1p-u 5.0%

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

      4. expm1-undefine 96.4%

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

   6. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 5.0%

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

   8. Simplified 5.0%

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

      1. expm1-undefine 96.4%

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

      2. flip-- 100.0%

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

      3. log1p-undefine 100.0%

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

      4. rem-exp-log 100.0%

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

      5. log1p-undefine 100.0%

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

      6. rem-exp-log 100.0%

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

      7. metadata-eval 100.0%

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

      8. log1p-undefine 100.0%

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

      9. rem-exp-log 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in l around 0 100.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.052:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(-1 + w \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(\ell + 1\right) \cdot \left(\ell + 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.4:\\ \;\;\;\;\ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.4) (+ l (* w (* 0.5 (* w l)))) (+ -1.0 (+ l 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.4) {
		tmp = l + (w * (0.5 * (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 <= 1.4d0) then
        tmp = l + (w * (0.5d0 * (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 <= 1.4) {
		tmp = l + (w * (0.5 * (w * l)));
	} else {
		tmp = -1.0 + (l + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.4:
		tmp = l + (w * (0.5 * (w * l)))
	else:
		tmp = -1.0 + (l + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.4)
		tmp = Float64(l + Float64(w * Float64(0.5 * 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 <= 1.4)
		tmp = l + (w * (0.5 * (w * l)));
	else
		tmp = -1.0 + (l + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.4], N[(l + N[(w * N[(0.5 * N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(l + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.3999999999999999

   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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   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 98.1%

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

   6. Taylor expanded in w around 0 80.8%

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

      1. associate-*r* 80.8%

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

      2. neg-mul-1 80.8%

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

      3. distribute-rgt-out 80.8%

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

      4. metadata-eval 80.8%

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

   8. Simplified 80.8%

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

   9. Taylor expanded in w around inf 80.8%

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

   if 1.3999999999999999 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

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

      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 0.0%

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

      7. sqr-neg 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. pow1 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%

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

      21. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 61.4%

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

      1. unpow1/3 61.4%

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

      2. rem-cbrt-cube 5.0%

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

      3. expm1-log1p-u 5.0%

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

      4. expm1-undefine 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 5.0%

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

   8. Simplified 5.0%

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

      1. expm1-undefine 100.0%

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

      2. log1p-undefine 100.0%

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

      3. rem-exp-log 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.4:\\ \;\;\;\;\ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.09:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.09) (- l (* w l)) (+ -1.0 (+ l 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.09) {
		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 <= 0.09d0) 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 <= 0.09) {
		tmp = l - (w * l);
	} else {
		tmp = -1.0 + (l + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.09:
		tmp = l - (w * l)
	else:
		tmp = -1.0 + (l + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.09)
		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 <= 0.09)
		tmp = l - (w * l);
	else
		tmp = -1.0 + (l + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.09], 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 < 0.089999999999999997

   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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   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 81.1%

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

      1. associate-*r* 81.1%

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

      2. neg-mul-1 81.1%

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

      3. distribute-rgt-out 81.1%

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

      4. metadata-eval 81.1%

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

   8. Simplified 81.1%

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

   9. Taylor expanded in w around 0 71.5%

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

       1. mul-1-neg 71.5%

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

       2. *-commutative 71.5%

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

       3. unsub-neg 71.5%

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

       4. *-commutative 71.5%

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

   11. Simplified 71.5%

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

   if 0.089999999999999997 < w

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

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

      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 3.8%

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

      7. sqr-neg 3.8%

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

      8. sqrt-unprod 3.8%

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

      9. add-sqr-sqrt 3.8%

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

      10. pow1 3.8%

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

      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%

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

      21. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 62.9%

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

      1. unpow1/3 62.9%

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

      2. rem-cbrt-cube 5.0%

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

      3. expm1-log1p-u 5.0%

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

      4. expm1-undefine 96.4%

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

   6. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 5.0%

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

   8. Simplified 5.0%

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

      1. expm1-undefine 96.4%

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

      2. log1p-undefine 96.4%

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

      3. rem-exp-log 96.4%

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

   10. Applied egg-rr 96.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.09:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\ell + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.4% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -410:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -410.0) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -410.0) {
		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 <= (-410.0d0)) 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 <= -410.0) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -410.0:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -410.0)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -410.0)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -410.0], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -410

   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 w around 0 100.0%

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

   6. Taylor expanded in w around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. neg-mul-1 53.5%

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

      3. distribute-rgt-out 53.5%

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

      4. metadata-eval 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in w around 0 27.9%

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

       1. mul-1-neg 27.9%

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

       2. *-commutative 27.9%

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

       3. unsub-neg 27.9%

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

       4. *-commutative 27.9%

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

   11. Simplified 27.9%

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

   12. Taylor expanded in w around inf 27.9%

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

       1. mul-1-neg 27.9%

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

   14. Simplified 27.9%

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

   if -410 < 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. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   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 97.4%

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

   6. Taylor expanded in w around 0 83.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -410:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% 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 99.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.3%

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

6. Taylor expanded in w around 0 73.1%

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

   1. associate-*r* 73.1%

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

   2. neg-mul-1 73.1%

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

   3. distribute-rgt-out 73.1%

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

   4. metadata-eval 73.1%

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

8. Simplified 73.1%

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

9. Taylor expanded in w around 0 64.6%

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

    1. mul-1-neg 64.6%

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

    2. *-commutative 64.6%

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

    3. unsub-neg 64.6%

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

    4. *-commutative 64.6%

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

11. Simplified 64.6%

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

12. Final simplification 64.6%

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

Alternative 11: 56.5% 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 99.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.3%

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

6. Taylor expanded in w around 0 56.7%

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

7. Final simplification 56.7%

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

Reproduce

herbie shell --seed 2024077 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))