exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 14.8s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- 0.0 w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp((0.0 - w)) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp((0.0d0 - w)) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp((0.0 - w)) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp((0.0 - w)) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(0.0 - w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp((0.0 - w)) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg N/A

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

   2. associate-*l/ N/A

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

   3. *-lft-identity N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

   7. exp-lowering-exp.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

3. Simplified 99.8%

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

Alternative 3: 97.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg N/A

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

   2. associate-*l/ N/A

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

   3. *-lft-identity N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

   7. exp-lowering-exp.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

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

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
6. Step-by-step derivation

7. Simplified 98.9%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]
8. Add Preprocessing

Alternative 4: 94.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;w \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.215:\\ \;\;\;\;\frac{\ell \cdot \left(1 - t\_0 \cdot t\_0\right)}{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666)))))))
   (if (<= w -1.02e+103)
     (* l (* w (* -0.16666666666666666 (* w w))))
     (if (<= w 0.215) (/ (* l (- 1.0 (* t_0 t_0))) (- 1.0 t_0)) 0.0))))

C

double code(double w, double l) {
	double t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))));
	double tmp;
	if (w <= -1.02e+103) {
		tmp = l * (w * (-0.16666666666666666 * (w * w)));
	} else if (w <= 0.215) {
		tmp = (l * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0)))))
    if (w <= (-1.02d+103)) then
        tmp = l * (w * ((-0.16666666666666666d0) * (w * w)))
    else if (w <= 0.215d0) then
        tmp = (l * (1.0d0 - (t_0 * t_0))) / (1.0d0 - t_0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))));
	double tmp;
	if (w <= -1.02e+103) {
		tmp = l * (w * (-0.16666666666666666 * (w * w)));
	} else if (w <= 0.215) {
		tmp = (l * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))
	tmp = 0
	if w <= -1.02e+103:
		tmp = l * (w * (-0.16666666666666666 * (w * w)))
	elif w <= 0.215:
		tmp = (l * (1.0 - (t_0 * t_0))) / (1.0 - t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))
	tmp = 0.0
	if (w <= -1.02e+103)
		tmp = Float64(l * Float64(w * Float64(-0.16666666666666666 * Float64(w * w))));
	elseif (w <= 0.215)
		tmp = Float64(Float64(l * Float64(1.0 - Float64(t_0 * t_0))) / Float64(1.0 - t_0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))));
	tmp = 0.0;
	if (w <= -1.02e+103)
		tmp = l * (w * (-0.16666666666666666 * (w * w)));
	elseif (w <= 0.215)
		tmp = (l * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(-1.0 + N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.02e+103], N[(l * N[(w * N[(-0.16666666666666666 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.215], N[(N[(l * N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if w < -1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot w\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 100.0%

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

   9. Taylor expanded in w around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot {w}^{3}\right)}, \ell\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({w}^{3} \cdot \frac{-1}{6}\right), \ell\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot \left(w \cdot w\right)\right) \cdot \frac{-1}{6}\right), \ell\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot {w}^{2}\right) \cdot \frac{-1}{6}\right), \ell\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left({w}^{2} \cdot \frac{-1}{6}\right)\right), \ell\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\frac{-1}{6} \cdot {w}^{2}\right)\right), \ell\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\frac{-1}{6} \cdot \left(w \cdot w\right)\right)\right), \ell\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\left(\frac{-1}{6} \cdot w\right) \cdot w\right)\right), \ell\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(\left(\frac{-1}{6} \cdot w\right) \cdot w\right)\right), \ell\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(\frac{-1}{6} \cdot \left(w \cdot w\right)\right)\right), \ell\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(\frac{-1}{6} \cdot {w}^{2}\right)\right), \ell\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\frac{-1}{6}, \left({w}^{2}\right)\right)\right), \ell\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\frac{-1}{6}, \left(w \cdot w\right)\right)\right), \ell\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, w\right)\right)\right), \ell\right) \]

   11. Simplified 100.0%

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

   if -1.01999999999999991e103 < w < 0.214999999999999997

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot w\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 82.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 82.7%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 84.1%

      \[\leadsto \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right) \cdot \left(w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right)}{1 - w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)} \cdot \ell \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(1 \cdot 1 - \left(w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right) \cdot \left(w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right)\right) \cdot \ell}{\color{blue}{1 - w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right) \cdot \left(w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right)\right) \cdot \ell\right), \color{blue}{\left(1 - w \cdot \left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right)}\right) \]

   10. Applied egg-rr 92.5%

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

   if 0.214999999999999997 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.215:\\ \;\;\;\;\frac{\ell \cdot \left(1 - \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\right)}{1 - w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + w \cdot -0.16666666666666666\\ t_1 := w \cdot t\_0\\ \mathbf{if}\;w \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(\left(w \cdot w\right) \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 0.19:\\ \;\;\;\;\ell \cdot \left(1 + \frac{w \cdot \left(1 - w \cdot \left(t\_0 \cdot t\_1\right)\right)}{-1 - t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* w -0.16666666666666666))) (t_1 (* w t_0)))
   (if (<= w -2e+154)
     (* l (* (* w w) 0.5))
     (if (<= w 0.19)
       (* l (+ 1.0 (/ (* w (- 1.0 (* w (* t_0 t_1)))) (- -1.0 t_1))))
       0.0))))

C

double code(double w, double l) {
	double t_0 = 0.5 + (w * -0.16666666666666666);
	double t_1 = w * t_0;
	double tmp;
	if (w <= -2e+154) {
		tmp = l * ((w * w) * 0.5);
	} else if (w <= 0.19) {
		tmp = l * (1.0 + ((w * (1.0 - (w * (t_0 * t_1)))) / (-1.0 - t_1)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 + (w * (-0.16666666666666666d0))
    t_1 = w * t_0
    if (w <= (-2d+154)) then
        tmp = l * ((w * w) * 0.5d0)
    else if (w <= 0.19d0) then
        tmp = l * (1.0d0 + ((w * (1.0d0 - (w * (t_0 * t_1)))) / ((-1.0d0) - t_1)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 0.5 + (w * -0.16666666666666666);
	double t_1 = w * t_0;
	double tmp;
	if (w <= -2e+154) {
		tmp = l * ((w * w) * 0.5);
	} else if (w <= 0.19) {
		tmp = l * (1.0 + ((w * (1.0 - (w * (t_0 * t_1)))) / (-1.0 - t_1)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 0.5 + (w * -0.16666666666666666)
	t_1 = w * t_0
	tmp = 0
	if w <= -2e+154:
		tmp = l * ((w * w) * 0.5)
	elif w <= 0.19:
		tmp = l * (1.0 + ((w * (1.0 - (w * (t_0 * t_1)))) / (-1.0 - t_1)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(0.5 + Float64(w * -0.16666666666666666))
	t_1 = Float64(w * t_0)
	tmp = 0.0
	if (w <= -2e+154)
		tmp = Float64(l * Float64(Float64(w * w) * 0.5));
	elseif (w <= 0.19)
		tmp = Float64(l * Float64(1.0 + Float64(Float64(w * Float64(1.0 - Float64(w * Float64(t_0 * t_1)))) / Float64(-1.0 - t_1))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 0.5 + (w * -0.16666666666666666);
	t_1 = w * t_0;
	tmp = 0.0;
	if (w <= -2e+154)
		tmp = l * ((w * w) * 0.5);
	elseif (w <= 0.19)
		tmp = l * (1.0 + ((w * (1.0 - (w * (t_0 * t_1)))) / (-1.0 - t_1)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(w * t$95$0), $MachinePrecision]}, If[LessEqual[w, -2e+154], N[(l * N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.19], N[(l * N[(1.0 + N[(N[(w * N[(1.0 - N[(w * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.00000000000000007e154

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \color{blue}{\left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(-1 \cdot \left(\left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right) \cdot w\right) - \ell\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) \cdot w - \ell\right)\right)\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\ell \cdot \left(-1 + \frac{1}{2}\right)\right)\right) \cdot w - \ell\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\ell \cdot \frac{-1}{2}\right)\right) \cdot w - \ell\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot \ell\right)\right) \cdot w - \ell\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\left(-1 \cdot \frac{-1}{2}\right) \cdot \ell\right) \cdot w - \ell\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot \ell\right) \cdot w - \ell\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot \ell\right) \cdot w\right), \color{blue}{\ell}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\left(\ell \cdot \frac{1}{2}\right) \cdot w\right), \ell\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\ell \cdot \left(\frac{1}{2} \cdot w\right)\right), \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w\right)\right), \ell\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(w \cdot \frac{1}{2}\right)\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(w, \frac{1}{2}\right)\right), \ell\right)\right)\right) \]

   10. Simplified 73.4%

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

   11. Taylor expanded in w around inf

       \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left({w}^{2} \cdot \left(-1 \cdot \frac{\ell}{w} + \frac{1}{2} \cdot \ell\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\left(-1 \cdot \frac{\ell}{w} + \frac{1}{2} \cdot \ell\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\color{blue}{-1 \cdot \frac{\ell}{w}} + \frac{1}{2} \cdot \ell\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\color{blue}{-1 \cdot \frac{\ell}{w}} + \frac{1}{2} \cdot \ell\right)\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-1 \cdot \ell}{w} + \color{blue}{\frac{1}{2}} \cdot \ell\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{\ell \cdot -1}{w} + \frac{1}{2} \cdot \ell\right)\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\ell \cdot \frac{-1}{w} + \color{blue}{\frac{1}{2}} \cdot \ell\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\ell \cdot \frac{-1}{w} + \ell \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       8. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\ell \cdot \color{blue}{\left(\frac{-1}{w} + \frac{1}{2}\right)}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{w} + \frac{1}{2}\right)}\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(\frac{-1}{w}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 100.0%

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, w\right), \frac{1}{2}\right)\right)\right)\right) \]

   13. Simplified 100.0%

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

   14. Taylor expanded in w around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} \cdot {w}^{2}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({w}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{2}, \left(w \cdot \color{blue}{w}\right)\right)\right) \]

       7. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right) \]

   16. Simplified 100.0%

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

   if -2.00000000000000007e154 < w < 0.19

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot w\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 84.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 84.0%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 85.3%

      \[\leadsto \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(-1 + w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right) \cdot w\right)\right), \ell\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1 \cdot -1 - \left(w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right) \cdot \left(w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)}{-1 - w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)} \cdot w\right)\right), \ell\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(-1 \cdot -1 - \left(w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right) \cdot \left(w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right) \cdot w}{-1 - w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)}\right)\right), \ell\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(-1 \cdot -1 - \left(w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right) \cdot \left(w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right) \cdot w\right), \left(-1 - w \cdot \left(\frac{1}{2} + w \cdot \frac{-1}{6}\right)\right)\right)\right), \ell\right) \]

   10. Applied egg-rr 87.9%

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

   if 0.19 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(\left(w \cdot w\right) \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 0.19:\\ \;\;\;\;\ell \cdot \left(1 + \frac{w \cdot \left(1 - w \cdot \left(\left(0.5 + w \cdot -0.16666666666666666\right) \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\right)}{-1 - w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.23:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.23)
   (* l (+ 1.0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666)))))))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.23) {
		tmp = l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.23d0) then
        tmp = l * (1.0d0 + (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0)))))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.23) {
		tmp = l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.23:
		tmp = l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.23)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.23)
		tmp = l * (1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.23], N[(l * N[(1.0 + N[(w * N[(-1.0 + N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.23000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot w\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 86.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 86.6%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 87.6%

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

   if 0.23000000000000001 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.23:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.3% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.056:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.23:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.056)
   (* l (* w (* -0.16666666666666666 (* w w))))
   (if (<= w 0.23) (/ l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.056) {
		tmp = l * (w * (-0.16666666666666666 * (w * w)));
	} else if (w <= 0.23) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.056d0)) then
        tmp = l * (w * ((-0.16666666666666666d0) * (w * w)))
    else if (w <= 0.23d0) then
        tmp = l / (w + 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.056) {
		tmp = l * (w * (-0.16666666666666666 * (w * w)));
	} else if (w <= 0.23) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.056:
		tmp = l * (w * (-0.16666666666666666 * (w * w)))
	elif w <= 0.23:
		tmp = l / (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.056)
		tmp = Float64(l * Float64(w * Float64(-0.16666666666666666 * Float64(w * w))));
	elseif (w <= 0.23)
		tmp = Float64(l / Float64(w + 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.056)
		tmp = l * (w * (-0.16666666666666666 * (w * w)));
	elseif (w <= 0.23)
		tmp = l / (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.056], N[(l * N[(w * N[(-0.16666666666666666 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.23], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.0560000000000000012

   1. Initial program 99.9%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot w\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 64.1%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 68.3%

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

   9. Taylor expanded in w around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot {w}^{3}\right)}, \ell\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({w}^{3} \cdot \frac{-1}{6}\right), \ell\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot \left(w \cdot w\right)\right) \cdot \frac{-1}{6}\right), \ell\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot {w}^{2}\right) \cdot \frac{-1}{6}\right), \ell\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left({w}^{2} \cdot \frac{-1}{6}\right)\right), \ell\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\frac{-1}{6} \cdot {w}^{2}\right)\right), \ell\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\frac{-1}{6} \cdot \left(w \cdot w\right)\right)\right), \ell\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\left(\frac{-1}{6} \cdot w\right) \cdot w\right)\right), \ell\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(\left(\frac{-1}{6} \cdot w\right) \cdot w\right)\right), \ell\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(\frac{-1}{6} \cdot \left(w \cdot w\right)\right)\right), \ell\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(\frac{-1}{6} \cdot {w}^{2}\right)\right), \ell\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\frac{-1}{6}, \left({w}^{2}\right)\right)\right), \ell\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\frac{-1}{6}, \left(w \cdot w\right)\right)\right), \ell\right) \]

       13. *-lowering-*.f64 68.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, w\right)\right)\right), \ell\right) \]

   11. Simplified 68.3%

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

   if -0.0560000000000000012 < w < 0.23000000000000001

   1. Initial program 99.6%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.6%

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

   8. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(1 + w\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\ell, \left(w + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(w, \color{blue}{1}\right)\right) \]

   10. Simplified 98.6%

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

   if 0.23000000000000001 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.056:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.23:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.052:\\ \;\;\;\;w \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.21:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.052)
   (* w (* -0.16666666666666666 (* l (* w w))))
   (if (<= w 0.21) (/ l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.052) {
		tmp = w * (-0.16666666666666666 * (l * (w * w)));
	} else if (w <= 0.21) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	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 = w * ((-0.16666666666666666d0) * (l * (w * w)))
    else if (w <= 0.21d0) then
        tmp = l / (w + 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.052) {
		tmp = w * (-0.16666666666666666 * (l * (w * w)));
	} else if (w <= 0.21) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.052:
		tmp = w * (-0.16666666666666666 * (l * (w * w)))
	elif w <= 0.21:
		tmp = l / (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.052)
		tmp = Float64(w * Float64(-0.16666666666666666 * Float64(l * Float64(w * w))));
	elseif (w <= 0.21)
		tmp = Float64(l / Float64(w + 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.052)
		tmp = w * (-0.16666666666666666 * (l * (w * w)));
	elseif (w <= 0.21)
		tmp = l / (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.052], N[(w * N[(-0.16666666666666666 * N[(l * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.21], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.0519999999999999976

   1. Initial program 99.9%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \left(-1 + w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot w\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 64.1%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\ell}\right) \]
   7. Step-by-step derivation

   8. Simplified 68.3%

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

   9. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left(\ell \cdot {w}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \ell\right) \cdot \color{blue}{{w}^{3}} \]

       2. unpow3 N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \ell\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{w}\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \ell\right) \cdot \left({w}^{2} \cdot w\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot \ell\right) \cdot {w}^{2}\right) \cdot \color{blue}{w} \]

       5. unpow2 N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot \ell\right) \cdot \left(w \cdot w\right)\right) \cdot w \]

       6. associate-*l* N/A

          \[\leadsto \left(\left(\left(\frac{-1}{6} \cdot \ell\right) \cdot w\right) \cdot w\right) \cdot w \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot \left(\ell \cdot w\right)\right) \cdot w\right) \cdot w \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{6} \cdot \left(\ell \cdot w\right)\right) \cdot w\right), \color{blue}{w}\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left(\left(\ell \cdot w\right) \cdot w\right)\right), w\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left(\ell \cdot \left(w \cdot w\right)\right)\right), w\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left(\ell \cdot {w}^{2}\right)\right), w\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(\ell \cdot {w}^{2}\right)\right), w\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\ell, \left({w}^{2}\right)\right)\right), w\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\ell, \left(w \cdot w\right)\right)\right), w\right) \]

       15. *-lowering-*.f64 63.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(w, w\right)\right)\right), w\right) \]

   11. Simplified 63.4%

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

   if -0.0519999999999999976 < w < 0.209999999999999992

   1. Initial program 99.6%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.6%

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

   8. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(1 + w\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\ell, \left(w + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(w, \color{blue}{1}\right)\right) \]

   10. Simplified 98.6%

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

   if 0.209999999999999992 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.052:\\ \;\;\;\;w \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.21:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.0% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.068:\\ \;\;\;\;\ell \cdot \left(\left(w \cdot w\right) \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 0.3:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.068) (* l (* (* w w) 0.5)) (if (<= w 0.3) (/ l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.068) {
		tmp = l * ((w * w) * 0.5);
	} else if (w <= 0.3) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.068d0)) then
        tmp = l * ((w * w) * 0.5d0)
    else if (w <= 0.3d0) then
        tmp = l / (w + 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.068) {
		tmp = l * ((w * w) * 0.5);
	} else if (w <= 0.3) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.068:
		tmp = l * ((w * w) * 0.5)
	elif w <= 0.3:
		tmp = l / (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.068)
		tmp = Float64(l * Float64(Float64(w * w) * 0.5));
	elseif (w <= 0.3)
		tmp = Float64(l / Float64(w + 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.068)
		tmp = l * ((w * w) * 0.5);
	elseif (w <= 0.3)
		tmp = l / (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.068], N[(l * N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.3], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.068000000000000005

   1. Initial program 99.9%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \color{blue}{\left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(-1 \cdot \left(\left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right) \cdot w\right) - \ell\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) \cdot w - \ell\right)\right)\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\ell \cdot \left(-1 + \frac{1}{2}\right)\right)\right) \cdot w - \ell\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\ell \cdot \frac{-1}{2}\right)\right) \cdot w - \ell\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot \ell\right)\right) \cdot w - \ell\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\left(-1 \cdot \frac{-1}{2}\right) \cdot \ell\right) \cdot w - \ell\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot \ell\right) \cdot w - \ell\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot \ell\right) \cdot w\right), \color{blue}{\ell}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\left(\ell \cdot \frac{1}{2}\right) \cdot w\right), \ell\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\ell \cdot \left(\frac{1}{2} \cdot w\right)\right), \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w\right)\right), \ell\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(w \cdot \frac{1}{2}\right)\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 45.2%

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(w, \frac{1}{2}\right)\right), \ell\right)\right)\right) \]

   10. Simplified 45.2%

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

   11. Taylor expanded in w around inf

       \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left({w}^{2} \cdot \left(-1 \cdot \frac{\ell}{w} + \frac{1}{2} \cdot \ell\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\left(-1 \cdot \frac{\ell}{w} + \frac{1}{2} \cdot \ell\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\color{blue}{-1 \cdot \frac{\ell}{w}} + \frac{1}{2} \cdot \ell\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\color{blue}{-1 \cdot \frac{\ell}{w}} + \frac{1}{2} \cdot \ell\right)\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-1 \cdot \ell}{w} + \color{blue}{\frac{1}{2}} \cdot \ell\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{\ell \cdot -1}{w} + \frac{1}{2} \cdot \ell\right)\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\ell \cdot \frac{-1}{w} + \color{blue}{\frac{1}{2}} \cdot \ell\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\ell \cdot \frac{-1}{w} + \ell \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       8. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\ell \cdot \color{blue}{\left(\frac{-1}{w} + \frac{1}{2}\right)}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{w} + \frac{1}{2}\right)}\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(\frac{-1}{w}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 57.2%

           \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, w\right), \frac{1}{2}\right)\right)\right)\right) \]

   13. Simplified 57.2%

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

   14. Taylor expanded in w around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} \cdot {w}^{2}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({w}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{2}, \left(w \cdot \color{blue}{w}\right)\right)\right) \]

       7. *-lowering-*.f64 57.2%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right) \]

   16. Simplified 57.2%

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

   if -0.068000000000000005 < w < 0.299999999999999989

   1. Initial program 99.6%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.6%

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

   8. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(1 + w\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\ell, \left(w + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(w, \color{blue}{1}\right)\right) \]

   10. Simplified 98.6%

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

   if 0.299999999999999989 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.068:\\ \;\;\;\;\ell \cdot \left(\left(w \cdot w\right) \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 0.3:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.0122:\\ \;\;\;\;\left(0 - w\right) \cdot \ell\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.0122) (* (- 0.0 w) l) (if (<= w 0.18) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.0122) {
		tmp = (0.0 - w) * l;
	} else if (w <= 0.18) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.0122d0)) then
        tmp = (0.0d0 - w) * l
    else if (w <= 0.18d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.0122) {
		tmp = (0.0 - w) * l;
	} else if (w <= 0.18) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.0122:
		tmp = (0.0 - w) * l
	elif w <= 0.18:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.0122)
		tmp = Float64(Float64(0.0 - w) * l);
	elseif (w <= 0.18)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.0122)
		tmp = (0.0 - w) * l;
	elseif (w <= 0.18)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.0122], N[(N[(0.0 - w), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[w, 0.18], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.0122000000000000008

   1. Initial program 99.9%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.8%

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

   8. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \color{blue}{\left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \color{blue}{\left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(-1 \cdot \left(\left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right) \cdot w\right) - \ell\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) \cdot w - \ell\right)\right)\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\ell \cdot \left(-1 + \frac{1}{2}\right)\right)\right) \cdot w - \ell\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\ell \cdot \frac{-1}{2}\right)\right) \cdot w - \ell\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot \ell\right)\right) \cdot w - \ell\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\left(-1 \cdot \frac{-1}{2}\right) \cdot \ell\right) \cdot w - \ell\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \left(\left(\frac{1}{2} \cdot \ell\right) \cdot w - \ell\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\left(\frac{1}{2} \cdot \ell\right) \cdot w\right), \color{blue}{\ell}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\left(\ell \cdot \frac{1}{2}\right) \cdot w\right), \ell\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\left(\ell \cdot \left(\frac{1}{2} \cdot w\right)\right), \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{1}{2} \cdot w\right)\right), \ell\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(w \cdot \frac{1}{2}\right)\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 45.2%

          \[\leadsto \mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(w, \frac{1}{2}\right)\right), \ell\right)\right)\right) \]

   10. Simplified 45.2%

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

   11. Taylor expanded in w around 0

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

       1. mul-1-neg N/A

          \[\leadsto \ell + \left(\mathsf{neg}\left(\ell \cdot w\right)\right) \]

       2. unsub-neg N/A

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

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\ell, \color{blue}{\left(\ell \cdot w\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\ell, \left(w \cdot \color{blue}{\ell}\right)\right) \]

       5. *-lowering-*.f64 27.7%

          \[\leadsto \mathsf{\_.f64}\left(\ell, \mathsf{*.f64}\left(w, \color{blue}{\ell}\right)\right) \]

   13. Simplified 27.7%

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

   14. Taylor expanded in w around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\ell \cdot w\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(w \cdot \ell\right) \]

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w, \color{blue}{\left(-1 \cdot \ell\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(w, \left(\mathsf{neg}\left(\ell\right)\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(w, \left(0 - \color{blue}{\ell}\right)\right) \]

       8. --lowering--.f64 27.7%

          \[\leadsto \mathsf{*.f64}\left(w, \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right) \]

   16. Simplified 27.7%

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

   if -0.0122000000000000008 < w < 0.17999999999999999

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 98.6%

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

   if 0.17999999999999999 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.0122:\\ \;\;\;\;\left(0 - w\right) \cdot \ell\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.8% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.18:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.18) (- l (* w l)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.18) {
		tmp = l - (w * l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.18d0) then
        tmp = l - (w * l)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.18) {
		tmp = l - (w * l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.18:
		tmp = l - (w * l)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.18)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.18)
		tmp = l - (w * l);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.18], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.17999999999999999

   1. Initial program 99.7%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(e^{w}\right)}\right), \color{blue}{\left(e^{w}\right)}\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \left(e^{w}\right)\right), \left(e^{\color{blue}{w}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \left(e^{w}\right)\right) \]

      7. exp-lowering-exp.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \mathsf{exp.f64}\left(w\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.7%

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

   8. Taylor expanded in w around 0

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

      1. mul-1-neg N/A

         \[\leadsto \ell + \left(\mathsf{neg}\left(\ell \cdot w\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\ell, \color{blue}{\left(\ell \cdot w\right)}\right) \]

      4. *-lowering-*.f64 73.0%

         \[\leadsto \mathsf{\_.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \color{blue}{w}\right)\right) \]

   10. Simplified 73.0%

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

   if 0.17999999999999999 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.18:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.0% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.235:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.235) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.235) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.235d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.235) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.235:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.235)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.235)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.235], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.23499999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 64.4%

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

   if 0.23499999999999999 < w

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 16.2% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 0.0)

C

double code(double w, double l) {
	return 0.0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 0.0d0
end function

Java

public static double code(double w, double l) {
	return 0.0;
}

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

function tmp = code(w, l)
	tmp = 0.0;
end

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 18.2%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))