exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 14.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 97.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{1}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (/ 1.0 (exp w))))

C

double code(double w, double l) {
	return l * (1.0 / exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 / exp(w))
end function

Java

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

Python

def code(w, l):
	return l * (1.0 / math.exp(w))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 / exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 / exp(w));
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 98.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{w}\right)\right), \ell\right) \]

   5. exp-lowering-exp.f64 98.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(w\right)\right), \ell\right) \]

9. Applied egg-rr 98.0%

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

10. Final simplification 98.0%

    \[\leadsto \ell \cdot \frac{1}{e^{w}} \]
11. Add Preprocessing

Alternative 4: 97.8% 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.0%

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

Alternative 5: 93.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - w \cdot 0.16666666666666666\\ \mathbf{if}\;w \leq -1 \cdot 10^{+156}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\ell \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.24:\\ \;\;\;\;\ell \cdot \left(\frac{w \cdot \left(-1 + w \cdot \left(w \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}{w \cdot t\_0 + 1} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- 0.5 (* w 0.16666666666666666))))
   (if (<= w -1e+156)
     (* -0.16666666666666666 (* l (* w (* w w))))
     (if (<= w 0.24)
       (*
        l
        (+ (/ (* w (+ -1.0 (* w (* w (* t_0 t_0))))) (+ (* w t_0) 1.0)) 1.0))
       0.0))))

C

double code(double w, double l) {
	double t_0 = 0.5 - (w * 0.16666666666666666);
	double tmp;
	if (w <= -1e+156) {
		tmp = -0.16666666666666666 * (l * (w * (w * w)));
	} else if (w <= 0.24) {
		tmp = l * (((w * (-1.0 + (w * (w * (t_0 * t_0))))) / ((w * t_0) + 1.0)) + 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 - (w * 0.16666666666666666d0)
    if (w <= (-1d+156)) then
        tmp = (-0.16666666666666666d0) * (l * (w * (w * w)))
    else if (w <= 0.24d0) then
        tmp = l * (((w * ((-1.0d0) + (w * (w * (t_0 * t_0))))) / ((w * t_0) + 1.0d0)) + 1.0d0)
    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 tmp;
	if (w <= -1e+156) {
		tmp = -0.16666666666666666 * (l * (w * (w * w)));
	} else if (w <= 0.24) {
		tmp = l * (((w * (-1.0 + (w * (w * (t_0 * t_0))))) / ((w * t_0) + 1.0)) + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 0.5 - (w * 0.16666666666666666)
	tmp = 0
	if w <= -1e+156:
		tmp = -0.16666666666666666 * (l * (w * (w * w)))
	elif w <= 0.24:
		tmp = l * (((w * (-1.0 + (w * (w * (t_0 * t_0))))) / ((w * t_0) + 1.0)) + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(0.5 - Float64(w * 0.16666666666666666))
	tmp = 0.0
	if (w <= -1e+156)
		tmp = Float64(-0.16666666666666666 * Float64(l * Float64(w * Float64(w * w))));
	elseif (w <= 0.24)
		tmp = Float64(l * Float64(Float64(Float64(w * Float64(-1.0 + Float64(w * Float64(w * Float64(t_0 * t_0))))) / Float64(Float64(w * t_0) + 1.0)) + 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 0.5 - (w * 0.16666666666666666);
	tmp = 0.0;
	if (w <= -1e+156)
		tmp = -0.16666666666666666 * (l * (w * (w * w)));
	elseif (w <= 0.24)
		tmp = l * (((w * (-1.0 + (w * (w * (t_0 * t_0))))) / ((w * t_0) + 1.0)) + 1.0);
	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]}, If[LessEqual[w, -1e+156], N[(-0.16666666666666666 * N[(l * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.24], N[(l * N[(N[(N[(w * N[(-1.0 + N[(w * N[(w * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if w < -9.9999999999999998e155

   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

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

   10. Simplified 95.8%

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

   11. Taylor expanded in w around inf

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

       1. cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. associate-*r/ N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

   14. Taylor expanded in w around inf

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

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

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

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

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 100.0%

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

   16. Simplified 100.0%

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

   if -9.9999999999999998e155 < w < 0.23999999999999999

   1. Initial program 99.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 97.6%

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

   8. Taylor expanded in w around 0

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

   10. Simplified 88.1%

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

       1. *-commutative N/A

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

       2. flip-- N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 63.1%

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

   13. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\ell \cdot \left(1 + \frac{w \cdot \left({w}^{2} \cdot {\left(\frac{1}{2} - \frac{1}{6} \cdot w\right)}^{2} - 1\right)}{1 + w \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot w\right)}\right)} \]
   14. Step-by-step derivation

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

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

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

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

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

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

   15. Simplified 93.6%

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

   if 0.23999999999999999 < 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 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1 \cdot 10^{+156}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\ell \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.24:\\ \;\;\;\;\ell \cdot \left(\frac{w \cdot \left(-1 + w \cdot \left(w \cdot \left(\left(0.5 - w \cdot 0.16666666666666666\right) \cdot \left(0.5 - w \cdot 0.16666666666666666\right)\right)\right)\right)}{w \cdot \left(0.5 - w \cdot 0.16666666666666666\right) + 1} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -8.5 \cdot 10^{-17}:\\ \;\;\;\;\left(w \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{\ell \cdot 0.5 + \frac{\frac{\ell}{w} - \ell}{w}}{w} - \ell \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;w \leq 0.21:\\ \;\;\;\;\ell + \ell \cdot \left(\left(w \cdot w\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -8.5e-17)
   (*
    (* w (* w w))
    (- (/ (+ (* l 0.5) (/ (- (/ l w) l) w)) w) (* l 0.16666666666666666)))
   (if (<= w 0.21) (+ l (* l (* (* w w) 0.5))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -8.5e-17) {
		tmp = (w * (w * w)) * ((((l * 0.5) + (((l / w) - l) / w)) / w) - (l * 0.16666666666666666));
	} else if (w <= 0.21) {
		tmp = l + (l * ((w * w) * 0.5));
	} 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 <= (-8.5d-17)) then
        tmp = (w * (w * w)) * ((((l * 0.5d0) + (((l / w) - l) / w)) / w) - (l * 0.16666666666666666d0))
    else if (w <= 0.21d0) then
        tmp = l + (l * ((w * w) * 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -8.5e-17) {
		tmp = (w * (w * w)) * ((((l * 0.5) + (((l / w) - l) / w)) / w) - (l * 0.16666666666666666));
	} else if (w <= 0.21) {
		tmp = l + (l * ((w * w) * 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -8.5e-17:
		tmp = (w * (w * w)) * ((((l * 0.5) + (((l / w) - l) / w)) / w) - (l * 0.16666666666666666))
	elif w <= 0.21:
		tmp = l + (l * ((w * w) * 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -8.5e-17)
		tmp = Float64(Float64(w * Float64(w * w)) * Float64(Float64(Float64(Float64(l * 0.5) + Float64(Float64(Float64(l / w) - l) / w)) / w) - Float64(l * 0.16666666666666666)));
	elseif (w <= 0.21)
		tmp = Float64(l + Float64(l * Float64(Float64(w * w) * 0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -8.5e-17)
		tmp = (w * (w * w)) * ((((l * 0.5) + (((l / w) - l) / w)) / w) - (l * 0.16666666666666666));
	elseif (w <= 0.21)
		tmp = l + (l * ((w * w) * 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -8.5e-17], N[(N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l * 0.5), $MachinePrecision] + N[(N[(N[(l / w), $MachinePrecision] - l), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] - N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.21], N[(l + N[(l * N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -8.5e-17

   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 98.4%

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

   8. Taylor expanded in w around 0

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

   10. Simplified 76.6%

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

   11. Taylor expanded in w around -inf

       \[\leadsto \color{blue}{-1 \cdot \left({w}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w} + \frac{1}{6} \cdot \ell\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({w}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w} + \frac{1}{6} \cdot \ell\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{{w}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w} + \frac{1}{6} \cdot \ell\right)} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({w}^{3} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w} + \frac{1}{6} \cdot \ell\right)\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left({w}^{3}\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w} + \frac{1}{6} \cdot \ell\right)}\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(w \cdot \left(w \cdot w\right)\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}} + \frac{1}{6} \cdot \ell\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(w \cdot {w}^{2}\right), \left(-1 \cdot \color{blue}{\frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}} + \frac{1}{6} \cdot \ell\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left({w}^{2}\right)\right), \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}} + \frac{1}{6} \cdot \ell\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(w \cdot w\right)\right), \left(-1 \cdot \color{blue}{\frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}} + \frac{1}{6} \cdot \ell\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right), \left(-1 \cdot \color{blue}{\frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}} + \frac{1}{6} \cdot \ell\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{1}{6} \cdot \ell + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}}\right)\right)\right) \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{1}{6} \cdot \ell + \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}\right)\right)\right)\right)\right) \]

       12. unsub-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{1}{6} \cdot \ell - \color{blue}{\frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{\_.f64}\left(\left(\frac{1}{6} \cdot \ell\right), \color{blue}{\left(\frac{-1 \cdot \frac{\ell + -1 \cdot \frac{\ell}{w}}{w} + \frac{1}{2} \cdot \ell}{w}\right)}\right)\right)\right) \]

   13. Simplified 84.5%

       \[\leadsto \color{blue}{0 - \left(w \cdot \left(w \cdot w\right)\right) \cdot \left(\ell \cdot 0.16666666666666666 - \frac{\ell \cdot 0.5 - \frac{\ell - \frac{\ell}{w}}{w}}{w}\right)} \]

   if -8.5e-17 < w < 0.209999999999999992

   1. Initial program 99.5%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 97.9%

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

   8. Taylor expanded in w around 0

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

   10. Simplified 97.9%

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

   11. Taylor expanded in w around inf

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

       1. cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. associate-*r/ N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 72.0%

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

   13. Simplified 72.0%

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

   14. Taylor expanded in w around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.9%

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

   16. Simplified 97.9%

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

   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 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -8.5 \cdot 10^{-17}:\\ \;\;\;\;\left(w \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{\ell \cdot 0.5 + \frac{\frac{\ell}{w} - \ell}{w}}{w} - \ell \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;w \leq 0.21:\\ \;\;\;\;\ell + \ell \cdot \left(\left(w \cdot w\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.96:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\ell \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\ell \cdot \frac{-1}{-1 - w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.96)
   (* -0.16666666666666666 (* l (* w (* w w))))
   (if (<= w 0.18) (* l (/ -1.0 (- -1.0 w))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.96) {
		tmp = -0.16666666666666666 * (l * (w * (w * w)));
	} else if (w <= 0.18) {
		tmp = l * (-1.0 / (-1.0 - w));
	} 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 <= (-0.96d0)) then
        tmp = (-0.16666666666666666d0) * (l * (w * (w * w)))
    else if (w <= 0.18d0) then
        tmp = l * ((-1.0d0) / ((-1.0d0) - w))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.96) {
		tmp = -0.16666666666666666 * (l * (w * (w * w)));
	} else if (w <= 0.18) {
		tmp = l * (-1.0 / (-1.0 - w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.96:
		tmp = -0.16666666666666666 * (l * (w * (w * w)))
	elif w <= 0.18:
		tmp = l * (-1.0 / (-1.0 - w))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.96)
		tmp = Float64(-0.16666666666666666 * Float64(l * Float64(w * Float64(w * w))));
	elseif (w <= 0.18)
		tmp = Float64(l * Float64(-1.0 / Float64(-1.0 - w)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.96)
		tmp = -0.16666666666666666 * (l * (w * (w * w)));
	elseif (w <= 0.18)
		tmp = l * (-1.0 / (-1.0 - w));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.96], N[(-0.16666666666666666 * N[(l * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.18], N[(l * N[(-1.0 / N[(-1.0 - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.95999999999999996

   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

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

   10. Simplified 77.1%

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

   11. Taylor expanded in w around inf

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

       1. cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. associate-*r/ N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 79.4%

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

   13. Simplified 79.4%

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

   14. Taylor expanded in w around inf

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

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

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

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

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 85.4%

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

   16. Simplified 85.4%

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

   if -0.95999999999999996 < w < 0.17999999999999999

   1. Initial program 99.3%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 97.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(e^{w}\right)\right), \ell\right) \]

      5. exp-lowering-exp.f64 97.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(w\right)\right), \ell\right) \]

   9. Applied egg-rr 97.0%

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

   10. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 97.0%

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

   12. Simplified 97.0%

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

   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 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.96:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\ell \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\ell \cdot \frac{-1}{-1 - w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.135:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.135) (* l (- 1.0 w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.135) {
		tmp = l * (1.0 - w);
	} 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 <= 0.135d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.135) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.135:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.135)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.135)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.135], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.13500000000000001

   1. Initial program 99.5%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 98.1%

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

   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. *-rgt-identity N/A

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

      4. distribute-lft-out-- N/A

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

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

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

      6. --lowering--.f64 73.5%

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

   10. Simplified 73.5%

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

   if 0.13500000000000001 < 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 9: 70.0% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.145:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.145) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.145) {
		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 <= 0.145d0) 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 <= 0.145) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.145:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.145)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.145)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.145], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.14499999999999999

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 62.9%

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

   if 0.14499999999999999 < 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 10: 16.4% 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 99.6%

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

4. Applied egg-rr 18.5%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))