exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 19.4s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. w = 9.763154157186523e+44
  2. l = -6.498015975928506e-251

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in w around inf

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

   1. *-commutative N/A

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

   2. exp-to-pow N/A

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

   3. remove-double-neg N/A

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

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

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

   5. log-rec N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. +-rgt-identity N/A

      \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

   9. exp-sum N/A

      \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

   10. +-rgt-identity N/A

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

   11. unsub-neg N/A

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

   12. div-exp N/A

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

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

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

5. Simplified 99.7%

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

Alternative 3: 99.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.6%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 99.6%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right)\right)}\right) \]
   7. 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%

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

   8. 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 4: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -220:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -220.0)
   (/ l (exp w))
   (/ (pow l (exp w)) (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -220.0) {
		tmp = l / exp(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 <= (-220.0d0)) then
        tmp = l / exp(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 <= -220.0) {
		tmp = l / Math.exp(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 <= -220.0:
		tmp = l / math.exp(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 <= -220.0)
		tmp = Float64(l / exp(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 <= -220.0)
		tmp = l / exp(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, -220.0], N[(l / N[Exp[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 < -220

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 100.0%

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

   if -220 < w

   1. Initial program 99.6%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 99.6%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}\right) \]
   7. 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 99.0%

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

   8. Simplified 99.0%

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

Alternative 5: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 100.0%

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

   if -1 < w

   1. Initial program 99.6%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 99.6%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right), \color{blue}{\left(1 + w\right)}\right) \]
   7. 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 98.6%

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

   8. Simplified 98.6%

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

Alternative 6: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -310:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot \left(1 - w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -310.0) (/ l (exp w)) (* (pow l (exp w)) (- 1.0 w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -310.0) {
		tmp = l / exp(w);
	} else {
		tmp = pow(l, exp(w)) * (1.0 - 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 <= (-310.0d0)) then
        tmp = l / exp(w)
    else
        tmp = (l ** exp(w)) * (1.0d0 - w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -310.0) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) * (1.0 - w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -310.0:
		tmp = l / math.exp(w)
	else:
		tmp = math.pow(l, math.exp(w)) * (1.0 - w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -310.0)
		tmp = Float64(l / exp(w));
	else
		tmp = Float64((l ^ exp(w)) * Float64(1.0 - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -310.0)
		tmp = l / exp(w);
	else
		tmp = (l ^ exp(w)) * (1.0 - w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -310.0], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -310

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

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

      11. unsub-neg N/A

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

      12. div-exp N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 100.0%

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

   if -310 < w

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -310:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot \left(1 - w\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 97.1%

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

6. Final simplification 97.1%

   \[\leadsto e^{0 - w} \cdot \ell \]
7. Add Preprocessing

Alternative 8: 97.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in w around inf

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

   1. *-commutative N/A

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

   2. exp-to-pow N/A

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

   3. remove-double-neg N/A

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

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

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

   5. log-rec N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. +-rgt-identity N/A

      \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

   9. exp-sum N/A

      \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

   10. +-rgt-identity N/A

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

   11. unsub-neg N/A

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

   12. div-exp N/A

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

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

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

5. Simplified 99.7%

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

6. Taylor expanded in w around 0

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

8. Simplified 97.1%

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

Alternative 9: 90.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.18:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.18)
   (* 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 <= 0.18) {
		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 <= 0.18d0) 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 <= 0.18) {
		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 <= 0.18:
		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 <= 0.18)
		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 <= 0.18)
		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, 0.18], 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 < 0.17999999999999999

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.5%

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

   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 89.2%

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

   8. Simplified 89.2%

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

   if 0.17999999999999999 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.18:\\ \;\;\;\;\ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.7% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.13) (* l (+ 1.0 (* w (+ (* w 0.5) -1.0)))) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.13d0) then
        tmp = l * (1.0d0 + (w * ((w * 0.5d0) + (-1.0d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.13) {
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.13:
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)))
	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(Float64(w * 0.5) + -1.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.13)
		tmp = l * (1.0 + (w * ((w * 0.5) + -1.0)));
	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[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.13

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.5%

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

   6. Taylor expanded in w around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 84.4%

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

   8. Simplified 84.4%

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

   if 0.13 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 86.6%

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

Alternative 11: 77.0% 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 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 97.5%

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

   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. sub-neg N/A

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

      3. --lowering--.f64 72.6%

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

   8. Simplified 72.6%

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

   if 0.12 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.3% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.26:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.26) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.26) {
		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 <= 0.26d0) 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 <= 0.26) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.26:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.26)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.26)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.26], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.26000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 64.9%

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

   if 0.26000000000000001 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

Alternative 13: 17.0% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

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

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.7%

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

   1. exp-neg N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. mul0-lft N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. mul0-lft N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

   14. metadata-eval N/A

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

   15. flip-- N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

   17. metadata-eval N/A

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

   18. associate-/r/ N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

   23. metadata-eval N/A

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

4. Applied egg-rr 17.1%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))