exp-w (used to crash)

Percentage Accurate: 99.4% → 99.5%

Time: 16.1s

Alternatives: 20

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: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\left(w + 1\right) + \left(0.5 + w \cdot 0.16666666666666666\right) \cdot \left(w \cdot w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (exp (- 0.0 w))
   (/
    (pow l (exp w))
    (+ (+ w 1.0) (* (+ 0.5 (* w 0.16666666666666666)) (* w w))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = exp((0.0 - w));
	} else {
		tmp = pow(l, exp(w)) / ((w + 1.0) + ((0.5 + (w * 0.16666666666666666)) * (w * w)));
	}
	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 = exp((0.0d0 - w))
    else
        tmp = (l ** exp(w)) / ((w + 1.0d0) + ((0.5d0 + (w * 0.16666666666666666d0)) * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = Math.pow(l, Math.exp(w)) / ((w + 1.0) + ((0.5 + (w * 0.16666666666666666)) * (w * w)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.6:
		tmp = math.exp((0.0 - w))
	else:
		tmp = math.pow(l, math.exp(w)) / ((w + 1.0) + ((0.5 + (w * 0.16666666666666666)) * (w * w)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[(w + 1.0), $MachinePrecision] + N[(N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(w * w), $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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -1.6000000000000001 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 99.3%

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

   7. Simplified 99.3%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Final simplification 98.6%

   \[\leadsto {\ell}^{\left(e^{w}\right)} \cdot e^{0 - w} \]
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 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.6%

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

3. Simplified 98.6%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (exp (- 0.0 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 = exp((0.0 - 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 = exp((0.0d0 - 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 = Math.exp((0.0 - 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 = math.exp((0.0 - 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 = exp(Float64(0.0 - 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 = exp((0.0 - 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[Exp[N[(0.0 - 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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -1.6000000000000001 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 99.3%

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

   7. Simplified 99.3%

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

Alternative 5: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.8:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3.8)
   (exp (- 0.0 w))
   (/ (pow l (exp w)) (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3.8) {
		tmp = exp((0.0 - w));
	} else {
		tmp = pow(l, exp(w)) / (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 <= (-3.8d0)) then
        tmp = exp((0.0d0 - w))
    else
        tmp = (l ** exp(w)) / (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 <= -3.8) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3.8:
		tmp = math.exp((0.0 - w))
	else:
		tmp = math.pow(l, math.exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3.8)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64((l ^ exp(w)) / 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 <= -3.8)
		tmp = exp((0.0 - w));
	else
		tmp = (l ^ exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3.8], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / 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 < -3.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -3.7999999999999998 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in w around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.2%

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

   7. Simplified 99.2%

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

Alternative 6: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666))))))))
   (if (<= w -1.6) (exp (- 0.0 w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.6) {
		tmp = exp((0.0 - w));
	} else {
		tmp = pow(l, t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0)))))
    if (w <= (-1.6d0)) then
        tmp = exp((0.0d0 - w))
    else
        tmp = (l ** t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.6) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = Math.pow(l, t_0) / t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))))
	tmp = 0
	if w <= -1.6:
		tmp = math.exp((0.0 - w))
	else:
		tmp = math.pow(l, t_0) / t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))))
	tmp = 0.0
	if (w <= -1.6)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666)))));
	tmp = 0.0;
	if (w <= -1.6)
		tmp = exp((0.0 - w));
	else
		tmp = (l ^ t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = 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, -1.6], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -1.6000000000000001 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in w around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.1%

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

   10. Simplified 99.1%

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

Alternative 7: 99.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = math.exp((0.0 - w))
	else:
		tmp = (w + -1.0) * ((l / (w + 1.0)) * (math.pow(l, (w * (1.0 + (w * 0.5)))) / (w + -1.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -1 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.0%

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

   10. Simplified 99.0%

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

       1. flip-+ N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       10. metadata-eval N/A

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

       11. sub-neg N/A

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

       12. metadata-eval N/A

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

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

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

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

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

       15. sub-neg N/A

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

       16. metadata-eval N/A

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

       17. +-lowering-+.f64 99.0%

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

   12. Applied egg-rr 99.0%

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

       1. unpow-prod-up N/A

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

       2. unpow1 N/A

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

       3. difference-of-sqr--1 N/A

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

       4. times-frac N/A

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

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

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       17. sub-neg N/A

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

       18. metadata-eval N/A

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

       19. +-lowering-+.f64 99.1%

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

   14. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

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

Alternative 8: 99.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = math.exp((0.0 - w))
	else:
		tmp = (l * math.pow(l, (w * (1.0 + (w * 0.5))))) / (w + 1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Power[l, N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -1 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.0%

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

   10. Simplified 99.0%

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

       1. +-commutative N/A

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

       2. pow-plus N/A

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 99.1%

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

   12. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

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

Alternative 9: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[(w + 1.0), $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. Add Preprocessing

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -1 < w

   1. Initial program 98.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   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 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 98.8%

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

   10. Simplified 98.8%

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

Alternative 10: 98.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7) (exp (- 0.0 w)) (if (<= w 0.18) (/ l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp((0.0 - w));
	} else if (w <= 0.18) {
		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 <= (-0.7d0)) then
        tmp = exp((0.0d0 - w))
    else if (w <= 0.18d0) 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 <= -0.7) {
		tmp = Math.exp((0.0 - w));
	} else if (w <= 0.18) {
		tmp = l / (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = math.exp((0.0 - w))
	elif w <= 0.18:
		tmp = l / (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(0.0 - w));
	elseif (w <= 0.18)
		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 <= -0.7)
		tmp = exp((0.0 - w));
	elseif (w <= 0.18)
		tmp = l / (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.18], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. prod-exp N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)\right), w\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell\right)\right) \cdot e^{w}\right)\right), w\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \ell \cdot e^{w}\right)\right)\right)\right), w\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      13. exp-lowering-exp.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.7%

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

   8. Simplified 99.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]

   if -0.69999999999999996 < w < 0.17999999999999999

   1. Initial program 99.7%

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

      1. exp-neg N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0

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

   7. Simplified 97.1%

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

   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 97.1%

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

   10. Simplified 97.1%

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

   if 0.17999999999999999 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

Alternative 11: 97.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
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 96.1%

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

Alternative 12: 94.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;w \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) + \frac{\left(w \cdot w\right) \cdot \left(0.25 - \left(w \cdot w\right) \cdot 0.027777777777777776\right)}{0.5 - w \cdot -0.16666666666666666}\right)\\ \mathbf{elif}\;w \leq 0.16:\\ \;\;\;\;\frac{\ell \cdot \left(1 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right)}{1 + t\_0 \cdot \left(-1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (+ -1.0 (* w (+ 0.5 (* w -0.16666666666666666)))))))
   (if (<= w -4.4e+51)
     (*
      l
      (+
       (- 1.0 w)
       (/
        (* (* w w) (- 0.25 (* (* w w) 0.027777777777777776)))
        (- 0.5 (* w -0.16666666666666666)))))
     (if (<= w 0.16)
       (/ (* l (+ 1.0 (* t_0 (* t_0 t_0)))) (+ 1.0 (* t_0 (+ -1.0 t_0))))
       0.0))))

C

double code(double w, double l) {
	double t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))));
	double tmp;
	if (w <= -4.4e+51) {
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))));
	} else if (w <= 0.16) {
		tmp = (l * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w * ((-1.0d0) + (w * (0.5d0 + (w * (-0.16666666666666666d0)))))
    if (w <= (-4.4d+51)) then
        tmp = l * ((1.0d0 - w) + (((w * w) * (0.25d0 - ((w * w) * 0.027777777777777776d0))) / (0.5d0 - (w * (-0.16666666666666666d0)))))
    else if (w <= 0.16d0) then
        tmp = (l * (1.0d0 + (t_0 * (t_0 * t_0)))) / (1.0d0 + (t_0 * ((-1.0d0) + t_0)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))));
	double tmp;
	if (w <= -4.4e+51) {
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))));
	} else if (w <= 0.16) {
		tmp = (l * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))))
	tmp = 0
	if w <= -4.4e+51:
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))))
	elif w <= 0.16:
		tmp = (l * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))))
	tmp = 0.0
	if (w <= -4.4e+51)
		tmp = Float64(l * Float64(Float64(1.0 - w) + Float64(Float64(Float64(w * w) * Float64(0.25 - Float64(Float64(w * w) * 0.027777777777777776))) / Float64(0.5 - Float64(w * -0.16666666666666666)))));
	elseif (w <= 0.16)
		tmp = Float64(Float64(l * Float64(1.0 + Float64(t_0 * Float64(t_0 * t_0)))) / Float64(1.0 + Float64(t_0 * Float64(-1.0 + t_0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = w * (-1.0 + (w * (0.5 + (w * -0.16666666666666666))));
	tmp = 0.0;
	if (w <= -4.4e+51)
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))));
	elseif (w <= 0.16)
		tmp = (l * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(-1.0 + N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4.4e+51], N[(l * N[(N[(1.0 - w), $MachinePrecision] + N[(N[(N[(w * w), $MachinePrecision] * N[(0.25 - N[(N[(w * w), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.16], N[(N[(l * N[(1.0 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if w < -4.39999999999999984e51

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 90.5%

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

   8. Simplified 90.5%

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. *-lowering-*.f64 90.5%

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

   10. Applied egg-rr 90.5%

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

       1. flip-+ N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} - \left(w \cdot \frac{-1}{6}\right) \cdot \left(w \cdot \frac{-1}{6}\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(w \cdot \frac{-1}{6}\right) \cdot \left(w \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

       7. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(w \cdot w\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right), \ell\right) \]

       13. *-lowering-*.f64 96.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right), \ell\right) \]

   12. Applied egg-rr 96.2%

       \[\leadsto \left(\left(1 - w\right) + \color{blue}{\frac{\left(0.25 - \left(w \cdot w\right) \cdot 0.027777777777777776\right) \cdot \left(w \cdot w\right)}{0.5 - w \cdot -0.16666666666666666}}\right) \cdot \ell \]

   if -4.39999999999999984e51 < w < 0.160000000000000003

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 95.6%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.1%

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

   8. Simplified 89.1%

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

      1. flip3-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 93.8%

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

   if 0.160000000000000003 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) + \frac{\left(w \cdot w\right) \cdot \left(0.25 - \left(w \cdot w\right) \cdot 0.027777777777777776\right)}{0.5 - w \cdot -0.16666666666666666}\right)\\ \mathbf{elif}\;w \leq 0.16:\\ \;\;\;\;\frac{\ell \cdot \left(1 + \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \left(\left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\right)\right)}{1 + \left(w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right) \cdot \left(-1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.105:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) + \frac{\left(w \cdot w\right) \cdot \left(0.25 - \left(w \cdot w\right) \cdot 0.027777777777777776\right)}{0.5 - w \cdot -0.16666666666666666}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.105)
   (*
    l
    (+
     (- 1.0 w)
     (/
      (* (* w w) (- 0.25 (* (* w w) 0.027777777777777776)))
      (- 0.5 (* w -0.16666666666666666)))))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.105) {
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (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 <= 0.105d0) then
        tmp = l * ((1.0d0 - w) + (((w * w) * (0.25d0 - ((w * w) * 0.027777777777777776d0))) / (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 <= 0.105) {
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.105:
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.105)
		tmp = Float64(l * Float64(Float64(1.0 - w) + Float64(Float64(Float64(w * w) * Float64(0.25 - Float64(Float64(w * w) * 0.027777777777777776))) / 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 <= 0.105)
		tmp = l * ((1.0 - w) + (((w * w) * (0.25 - ((w * w) * 0.027777777777777776))) / (0.5 - (w * -0.16666666666666666))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.105], N[(l * N[(N[(1.0 - w), $MachinePrecision] + N[(N[(N[(w * w), $MachinePrecision] * N[(0.25 - N[(N[(w * w), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.104999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 96.7%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.4%

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

   8. Simplified 89.4%

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. *-lowering-*.f64 89.4%

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

   10. Applied egg-rr 89.4%

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

       1. flip-+ N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} - \left(w \cdot \frac{-1}{6}\right) \cdot \left(w \cdot \frac{-1}{6}\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(w \cdot \frac{-1}{6}\right) \cdot \left(w \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

       7. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(w \cdot w\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \left(w \cdot w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{1}{2} - w \cdot \frac{-1}{6}\right)\right)\right), \ell\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(w \cdot \frac{-1}{6}\right)\right)\right)\right), \ell\right) \]

       13. *-lowering-*.f64 90.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{36}\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(w, \frac{-1}{6}\right)\right)\right)\right), \ell\right) \]

   12. Applied egg-rr 90.8%

       \[\leadsto \left(\left(1 - w\right) + \color{blue}{\frac{\left(0.25 - \left(w \cdot w\right) \cdot 0.027777777777777776\right) \cdot \left(w \cdot w\right)}{0.5 - w \cdot -0.16666666666666666}}\right) \cdot \ell \]

   if 0.104999999999999996 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.105:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) + \frac{\left(w \cdot w\right) \cdot \left(0.25 - \left(w \cdot w\right) \cdot 0.027777777777777776\right)}{0.5 - w \cdot -0.16666666666666666}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.13:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.13)
   (* 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 <= 0.13) {
		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 <= 0.13d0) 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 <= 0.13) {
		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 <= 0.13:
		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 <= 0.13)
		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 <= 0.13)
		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, 0.13], 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 < 0.13

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 96.7%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.4%

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

   8. Simplified 89.4%

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

   if 0.13 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.13:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.3% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.145:\\ \;\;\;\;\ell \cdot \left(1 + \left(w \cdot w\right) \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.145)
   (* l (+ 1.0 (* (* w w) (+ 0.5 (* w -0.16666666666666666)))))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.145) {
		tmp = l * (1.0 + ((w * 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 <= 0.145d0) then
        tmp = l * (1.0d0 + ((w * 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 <= 0.145) {
		tmp = l * (1.0 + ((w * w) * (0.5 + (w * -0.16666666666666666))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.145:
		tmp = l * (1.0 + ((w * w) * (0.5 + (w * -0.16666666666666666))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.145)
		tmp = Float64(l * Float64(1.0 + Float64(Float64(w * 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 <= 0.145)
		tmp = l * (1.0 + ((w * w) * (0.5 + (w * -0.16666666666666666))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.145], N[(l * N[(1.0 + N[(N[(w * w), $MachinePrecision] * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.14499999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 96.7%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.4%

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

   8. Simplified 89.4%

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. *-lowering-*.f64 89.4%

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

   10. Applied egg-rr 89.4%

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

   11. Taylor expanded in w around 0

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

   13. Simplified 89.4%

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

   if 0.14499999999999999 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.145:\\ \;\;\;\;\ell \cdot \left(1 + \left(w \cdot w\right) \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(\left(w \cdot w\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.17) (* l (+ 1.0 (* w (* (* w w) -0.16666666666666666)))) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l * (1.0 + (w * ((w * 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 <= 0.17d0) then
        tmp = l * (1.0d0 + (w * ((w * 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 <= 0.17) {
		tmp = l * (1.0 + (w * ((w * w) * -0.16666666666666666)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.17:
		tmp = l * (1.0 + (w * ((w * w) * -0.16666666666666666)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.17)
		tmp = Float64(l * Float64(1.0 + Float64(w * Float64(Float64(w * w) * -0.16666666666666666))));
	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 * ((w * w) * -0.16666666666666666)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.17], N[(l * N[(1.0 + N[(w * N[(N[(w * w), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.170000000000000012

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 96.7%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.4%

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

   8. Simplified 89.4%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

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

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

       9. associate-*r* N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 89.4%

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

   11. Simplified 89.4%

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

   if 0.170000000000000012 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(\left(w \cdot w\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 91.3% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -190:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(\left(w \cdot w\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq 0.11:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -190.0)
   (* l (* w (* (* w w) -0.16666666666666666)))
   (if (<= w 0.11) (* l (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -190.0) {
		tmp = l * (w * ((w * w) * -0.16666666666666666));
	} else if (w <= 0.11) {
		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 <= (-190.0d0)) then
        tmp = l * (w * ((w * w) * (-0.16666666666666666d0)))
    else if (w <= 0.11d0) 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 <= -190.0) {
		tmp = l * (w * ((w * w) * -0.16666666666666666));
	} else if (w <= 0.11) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -190.0:
		tmp = l * (w * ((w * w) * -0.16666666666666666))
	elif w <= 0.11:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -190.0)
		tmp = Float64(l * Float64(w * Float64(Float64(w * w) * -0.16666666666666666)));
	elseif (w <= 0.11)
		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 <= -190.0)
		tmp = l * (w * ((w * w) * -0.16666666666666666));
	elseif (w <= 0.11)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -190.0], N[(l * N[(w * N[(N[(w * w), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.11], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -190

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 98.5%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 74.5%

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

   8. Simplified 74.5%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

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

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

       9. associate-*r* N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 74.5%

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

   11. Simplified 74.5%

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

   if -190 < 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

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

   5. Simplified 95.9%

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

   6. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 95.9%

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

   8. Simplified 95.9%

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

   if 0.110000000000000001 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -190:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(\left(w \cdot w\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;w \leq 0.11:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.8% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.12) (* l (- 1.0 w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.12) {
		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 <= 0.12d0) 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 <= 0.12) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.12:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.12)
		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 <= 0.12)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.12], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.12

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 96.7%

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

   6. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 74.9%

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

   8. Simplified 74.9%

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

   if 0.12 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 70.6% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.15:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.15) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.15) {
		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 <= 0.15d0) 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 <= 0.15) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.15:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.15)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.15)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.15], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.149999999999999994

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 67.9%

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

   if 0.149999999999999994 < w

   1. Initial program 93.3%

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

   4. Applied egg-rr 93.4%

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

Alternative 20: 16.8% 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 98.6%

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

4. Applied egg-rr 19.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))