exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 15.8s

Alternatives: 17

Speedup: 1.4×

• 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (/ 1.0 (exp w))
   (/
    (pow l (exp w))
    (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.6d0)) then
        tmp = 1.0d0 / exp(w)
    else
        tmp = (l ** exp(w)) / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.6:
		tmp = 1.0 / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.6)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = Float64((l ^ exp(w)) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.6)
		tmp = 1.0 / exp(w);
	else
		tmp = (l ^ exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.4%

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

   7. Simplified 99.4%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 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 99.3%

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

3. Final simplification 99.3%

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

Alternative 3: 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 99.3%

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

Alternative 4: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.9:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot \frac{1}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3.9)
   (/ 1.0 (exp w))
   (* (pow l (exp w)) (/ 1.0 (+ 1.0 (* w (+ 1.0 (* w 0.5))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3.9) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, exp(w)) * (1.0 / (1.0 + (w * (1.0 + (w * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-3.9d0)) then
        tmp = 1.0d0 / exp(w)
    else
        tmp = (l ** exp(w)) * (1.0d0 / (1.0d0 + (w * (1.0d0 + (w * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -3.9) {
		tmp = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) * (1.0 / (1.0 + (w * (1.0 + (w * 0.5)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3.9:
		tmp = 1.0 / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) * (1.0 / (1.0 + (w * (1.0 + (w * 0.5)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3.9)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = Float64((l ^ exp(w)) * Float64(1.0 / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -3.9)
		tmp = 1.0 / exp(w);
	else
		tmp = (l ^ exp(w)) * (1.0 / (1.0 + (w * (1.0 + (w * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3.9], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -3.89999999999999991

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -3.89999999999999991 < w

   1. Initial program 99.1%

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

   3. Taylor expanded in w around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(w\right)}\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. exp-neg N/A

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

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

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

      3. exp-lowering-exp.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.2%

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

   8. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{1 + w \cdot \left(1 + w \cdot 0.5\right)}} \cdot {\ell}^{\left(e^{w}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.9:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot \frac{1}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0) (/ 1.0 (exp w)) (/ (pow l (exp w)) (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, exp(w)) / (w + 1.0);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = 1.0 / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) / (w + 1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.1%

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

   7. Simplified 99.1%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{w + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (/ 1.0 (exp w))
   (pow l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.6d0)) then
        tmp = 1.0d0 / exp(w)
    else
        tmp = l ** (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.6:
		tmp = 1.0 / math.exp(w)
	else:
		tmp = math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.6)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = l ^ Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.6)
		tmp = 1.0 / exp(w);
	else
		tmp = l ^ (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Power[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]]

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in w around 0

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

   11. Simplified 98.5%

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

       1. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 98.5%

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

   13. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{{\ell}^{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(1 + w \cdot \left(1 + w \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3) (/ 1.0 (exp w)) (pow l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.3d0)) then
        tmp = 1.0d0 / exp(w)
    else
        tmp = l ** (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * 0.5)))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.3:
		tmp = 1.0 / math.exp(w)
	else:
		tmp = math.pow(l, (1.0 + (w * (1.0 + (w * 0.5)))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.3)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = l ^ Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.3)
		tmp = 1.0 / exp(w);
	else
		tmp = l ^ (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Power[l, N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.30000000000000004

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1.30000000000000004 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in w around 0

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

   11. Simplified 98.5%

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

   12. Taylor expanded in w around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

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

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

       6. *-lft-identity N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

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

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

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 98.5%

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

   14. Simplified 98.5%

       \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot 0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(1 + w \cdot \left(1 + w \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0) (/ 1.0 (exp w)) (pow l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, (w + 1.0));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, (w + 1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = 1.0 / math.exp(w)
	else:
		tmp = math.pow(l, (w + 1.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in w around 0

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

   11. Simplified 98.5%

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

   12. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 98.4%

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

   14. Simplified 98.4%

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

4. Final simplification 98.8%

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

Alternative 9: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.72:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{elif}\;w \leq 0.185:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.72) (/ 1.0 (exp w)) (if (<= w 0.185) (/ l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.72) {
		tmp = 1.0 / exp(w);
	} else if (w <= 0.185) {
		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 <= (-0.72d0)) then
        tmp = 1.0d0 / exp(w)
    else if (w <= 0.185d0) 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 <= -0.72) {
		tmp = 1.0 / Math.exp(w);
	} else if (w <= 0.185) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.72:
		tmp = 1.0 / math.exp(w)
	elif w <= 0.185:
		tmp = l / (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.72)
		tmp = Float64(1.0 / exp(w));
	elseif (w <= 0.185)
		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 <= -0.72)
		tmp = 1.0 / exp(w);
	elseif (w <= 0.185)
		tmp = l / (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.72], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.185], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.71999999999999997

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -0.71999999999999997 < w < 0.185

   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 97.6%

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

   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 97.6%

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

   10. Simplified 97.6%

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

   if 0.185 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

Alternative 10: 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 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 97.1%

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

Alternative 11: 90.0% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.2 \cdot 10^{+110}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.2e+110)
   (+ 1.0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666))))))
   (if (<= w 0.115) (+ l (* w (* l (+ (* w 0.5) -1.0)))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.2e+110) {
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	} else if (w <= 0.115) {
		tmp = l + (w * (l * ((w * 0.5) + -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 <= (-2.2d+110)) then
        tmp = 1.0d0 + (w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0))))))
    else if (w <= 0.115d0) then
        tmp = l + (w * (l * ((w * 0.5d0) + (-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 <= -2.2e+110) {
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	} else if (w <= 0.115) {
		tmp = l + (w * (l * ((w * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.2e+110:
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))))
	elif w <= 0.115:
		tmp = l + (w * (l * ((w * 0.5) + -1.0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.2e+110)
		tmp = Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))))));
	elseif (w <= 0.115)
		tmp = Float64(l + Float64(w * Float64(l * Float64(Float64(w * 0.5) + -1.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -2.2e+110)
		tmp = 1.0 + (w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666)))));
	elseif (w <= 0.115)
		tmp = l + (w * (l * ((w * 0.5) + -1.0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.2e+110], N[(1.0 + N[(w * N[(-1.0 + N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.115], N[(l + N[(w * N[(l * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -2.19999999999999992e110

   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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{\left(0 - 0\right)}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{0}\right), \mathsf{exp.f64}\left(w\right)\right) \]

      12. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

   if -2.19999999999999992e110 < w < 0.115000000000000005

   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 97.4%

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

   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. sub-neg N/A

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

      4. mul-1-neg N/A

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

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

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

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

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 86.3%

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

   10. Simplified 86.3%

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

   if 0.115000000000000005 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.2 \cdot 10^{+110}:\\ \;\;\;\;1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.11:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.11)
   (* 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 <= 0.11) {
		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 <= 0.11d0) 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 <= 0.11) {
		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 <= 0.11:
		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 <= 0.11)
		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 <= 0.11)
		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, 0.11], 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 < 0.110000000000000001

   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 88.8%

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

   5. Simplified 88.8%

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

   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 89.0%

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

   if 0.110000000000000001 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.11:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right)\right)\\ \mathbf{elif}\;w \leq 0.085:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.0) (* l (* w (* w 0.5))) (if (<= w 0.085) (/ l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = l * (w * (w * 0.5));
	} else if (w <= 0.085) {
		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 <= (-1.0d0)) then
        tmp = l * (w * (w * 0.5d0))
    else if (w <= 0.085d0) 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 <= -1.0) {
		tmp = l * (w * (w * 0.5));
	} else if (w <= 0.085) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = l * (w * (w * 0.5))
	elif w <= 0.085:
		tmp = l / (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = Float64(l * Float64(w * Float64(w * 0.5)));
	elseif (w <= 0.085)
		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 <= -1.0)
		tmp = l * (w * (w * 0.5));
	elseif (w <= 0.085)
		tmp = l / (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[(l * N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.085], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -1

   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 98.6%

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

      1. clear-num N/A

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

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

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

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

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

      4. exp-lowering-exp.f64 98.6%

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

   9. Applied egg-rr 98.6%

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

   10. Taylor expanded in w around 0

       \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \ell + \frac{1}{2} \cdot \ell\right)\right) - \ell\right)} \]
   11. 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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 51.5%

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

   12. Simplified 51.5%

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

   13. Taylor expanded in w around inf

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

       1. associate-*l* 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. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 60.9%

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

   15. Simplified 60.9%

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

   if -1 < w < 0.0850000000000000061

   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 97.6%

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

   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 97.6%

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

   10. Simplified 97.6%

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

   if 0.0850000000000000061 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

Alternative 14: 77.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.115) (- l (* w l)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.115) {
		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 <= 0.115d0) 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 <= 0.115) {
		tmp = l - (w * l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.115:
		tmp = l - (w * l)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.115)
		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 <= 0.115)
		tmp = l - (w * l);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.115], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.115000000000000005

   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 97.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. exp-lowering-exp.f64 97.8%

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

   9. Applied egg-rr 97.8%

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

   10. Taylor expanded in w around 0

       \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   11. 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 74.8%

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

   12. Simplified 74.8%

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

   if 0.115000000000000005 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.17) (* l (- 1.0 w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		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 <= 0.17d0) 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 <= 0.17) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.17:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.17)
		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 <= 0.17)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.17], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.170000000000000012

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 74.8%

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

   if 0.170000000000000012 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.8% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.25:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.25) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.25) {
		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 <= 0.25d0) 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 <= 0.25) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.25:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.25)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.25)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.25], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.25

   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 67.6%

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

   if 0.25 < w

   1. Initial program 97.5%

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

   4. Applied egg-rr 95.1%

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

Alternative 17: 16.5% 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 99.3%

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

4. Applied egg-rr 17.6%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))