exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 22.3s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. w = 2.1086363725849157e-79
  2. l = -2.723353739083481e+205

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: 99.5% 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, 1.0× 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: 88.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], 0.0, If[LessEqual[t$95$0, Infinity], N[(l * N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(l * N[(w * N[(w * N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 0.0

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 86.7

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

   8. Simplified 86.7%

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

   if +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 98.2%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. +-commutative N/A

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

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

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

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

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. accelerator-lowering-fma.f64 76.3

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

   8. Simplified 76.3%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. metadata-eval N/A

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

       12. associate-*l* N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f64 23.3

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

   11. Simplified 23.3%

       \[\leadsto \color{blue}{\ell \cdot \left(w \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq \infty:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 0.0

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 89.3

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

   8. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

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

Alternative 4: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \ell \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right)\right), \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- 0.0 w)) (pow l (exp w))) 0.0)
   0.0
   (fma w (* l (* w (* w -0.16666666666666666))) l)))

C

double code(double w, double l) {
	double tmp;
	if ((exp((0.0 - w)) * pow(l, exp(w))) <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = fma(w, (l * (w * (w * -0.16666666666666666))), l);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(0.0 - w)) * (l ^ exp(w))) <= 0.0)
		tmp = 0.0;
	else
		tmp = fma(w, Float64(l * Float64(w * Float64(w * -0.16666666666666666))), l);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 0.0

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. +-commutative N/A

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

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

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

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

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. accelerator-lowering-fma.f64 87.6

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

   8. Simplified 87.6%

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

      1. flip-+ N/A

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

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

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

      3. sub-neg N/A

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

      4. swap-sqr N/A

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

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

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(w, \left(\ell \cdot w\right) \cdot \frac{\mathsf{fma}\left(w \cdot w, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}{w \cdot \frac{-1}{6} - \frac{1}{2}} - \ell, \ell\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(w, \left(\ell \cdot w\right) \cdot \frac{\mathsf{fma}\left(w \cdot w, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)}{w \cdot \frac{-1}{6} - \frac{1}{2}} - \ell, \ell\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(w, \left(\ell \cdot w\right) \cdot \frac{\mathsf{fma}\left(w \cdot w, \frac{1}{36}, \color{blue}{\frac{-1}{4}}\right)}{w \cdot \frac{-1}{6} - \frac{1}{2}} - \ell, \ell\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(w, \left(\ell \cdot w\right) \cdot \frac{\mathsf{fma}\left(w \cdot w, \frac{1}{36}, \frac{-1}{4}\right)}{\color{blue}{w \cdot \frac{-1}{6} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} - \ell, \ell\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(w, \left(\ell \cdot w\right) \cdot \frac{\mathsf{fma}\left(w \cdot w, \frac{1}{36}, \frac{-1}{4}\right)}{\color{blue}{\mathsf{fma}\left(w, \frac{-1}{6}, \mathsf{neg}\left(\frac{1}{2}\right)\right)}} - \ell, \ell\right) \]

      12. metadata-eval 88.1

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

   10. Applied egg-rr 88.1%

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

   11. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 88.1

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

   13. Simplified 88.1%

       \[\leadsto \mathsf{fma}\left(w, \color{blue}{\ell \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right)\right)}, \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \ell \cdot \left(w \cdot \left(w \cdot -0.16666666666666666\right)\right), \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 0.0

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 86.7

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

   8. Simplified 86.7%

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

4. Final simplification 88.5%

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

Alternative 6: 71.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 0.0

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 66.7%

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

4. Final simplification 71.1%

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

Alternative 7: 97.8% accurate, 2.8× 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 0

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

5. Simplified 98.2%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

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

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

   5. exp-lowering-exp.f64 98.2

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

7. Applied egg-rr 98.2%

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

Alternative 8: 98.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.21:\\ \;\;\;\;\frac{\ell \cdot \left(1 - w \cdot w\right)}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (exp (- 0.0 w))
   (if (<= w 0.21) (/ (* l (- 1.0 (* w w))) (+ 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 <= 0.21) {
		tmp = (l * (1.0 - (w * w))) / (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 <= 0.21d0) then
        tmp = (l * (1.0d0 - (w * w))) / (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 <= 0.21) {
		tmp = (l * (1.0 - (w * w))) / (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 <= 0.21:
		tmp = (l * (1.0 - (w * w))) / (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 <= 0.21)
		tmp = Float64(Float64(l * Float64(1.0 - Float64(w * w))) / 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 <= 0.21)
		tmp = (l * (1.0 - (w * w))) / (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, 0.21], N[(N[(l * N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.8

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

   4. Applied egg-rr 98.8%

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

   if -0.69999999999999996 < w < 0.209999999999999992

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.5%

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

   6. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 97.5

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

   8. Simplified 97.5%

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

      1. flip-- N/A

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

      2. associate-*l/ N/A

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

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

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 97.5

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

   10. Applied egg-rr 97.5%

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

   if 0.209999999999999992 < 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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.21:\\ \;\;\;\;\frac{\ell \cdot \left(1 - w \cdot w\right)}{w + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 2.8× 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

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 10: 92.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.22:\\ \;\;\;\;\mathsf{fma}\left(w, \left(w \cdot \ell\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot 0.027777777777777776\right) \cdot \mathsf{fma}\left(w, -0.16666666666666666, -0.5\right) - \mathsf{fma}\left(w, -0.16666666666666666, -0.5\right) \cdot 0.25}{0.25} - \ell, \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.22)
   (fma
    w
    (-
     (*
      (* w l)
      (/
       (-
        (* (* (* w w) 0.027777777777777776) (fma w -0.16666666666666666 -0.5))
        (* (fma w -0.16666666666666666 -0.5) 0.25))
       0.25))
     l)
    l)
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.22) {
		tmp = fma(w, (((w * l) * (((((w * w) * 0.027777777777777776) * fma(w, -0.16666666666666666, -0.5)) - (fma(w, -0.16666666666666666, -0.5) * 0.25)) / 0.25)) - l), l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.22)
		tmp = fma(w, Float64(Float64(Float64(w * l) * Float64(Float64(Float64(Float64(Float64(w * w) * 0.027777777777777776) * fma(w, -0.16666666666666666, -0.5)) - Float64(fma(w, -0.16666666666666666, -0.5) * 0.25)) / 0.25)) - l), l);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.22], N[(w * N[(N[(N[(w * l), $MachinePrecision] * N[(N[(N[(N[(N[(w * w), $MachinePrecision] * 0.027777777777777776), $MachinePrecision] * N[(w * -0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(w * -0.16666666666666666 + -0.5), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision] + l), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.220000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. +-commutative N/A

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

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

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

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

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. accelerator-lowering-fma.f64 87.6

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

   8. Simplified 87.6%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. frac-sub N/A

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

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

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

   10. Applied egg-rr 74.0%

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

   11. Taylor expanded in w around 0

       \[\leadsto \mathsf{fma}\left(w, \left(\ell \cdot w\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \frac{1}{36}\right) \cdot \mathsf{fma}\left(w, \frac{-1}{6}, \frac{-1}{2}\right) - \mathsf{fma}\left(w, \frac{-1}{6}, \frac{-1}{2}\right) \cdot \frac{1}{4}}{\color{blue}{\frac{1}{4}}} - \ell, \ell\right) \]
   12. Step-by-step derivation

   13. Simplified 91.0%

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

   if 0.220000000000000001 < 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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.22:\\ \;\;\;\;\mathsf{fma}\left(w, \left(w \cdot \ell\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot 0.027777777777777776\right) \cdot \mathsf{fma}\left(w, -0.16666666666666666, -0.5\right) - \mathsf{fma}\left(w, -0.16666666666666666, -0.5\right) \cdot 0.25}{0.25} - \ell, \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.9% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.17d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < 0.170000000000000012

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.9%

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

   6. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.2

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

   8. Simplified 75.2%

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

   if 0.170000000000000012 < 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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 78.5%

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

Alternative 12: 16.0% accurate, 309.0× 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

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. mul0-lft N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. mul0-lft N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

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

   14. metadata-eval N/A

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

   15. flip-- N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. associate-/r/ N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

4. Applied egg-rr 16.0%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))