exp-w (used to crash)

Percentage Accurate: 99.4% → 99.3%

Time: 21.3s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. w = 2.755100279884728e-233
  2. l = -1.884430149918696e-290

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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.35 \cdot 10^{-14}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.35e-14) (exp (- (* (exp w) (log l)) w)) (pow l (exp w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.35e-14) {
		tmp = exp(((exp(w) * log(l)) - w));
	} else {
		tmp = pow(l, exp(w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.35d-14)) then
        tmp = exp(((exp(w) * log(l)) - w))
    else
        tmp = l ** exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.35e-14) {
		tmp = Math.exp(((Math.exp(w) * Math.log(l)) - w));
	} else {
		tmp = Math.pow(l, Math.exp(w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.35e-14:
		tmp = math.exp(((math.exp(w) * math.log(l)) - w))
	else:
		tmp = math.pow(l, math.exp(w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.35e-14)
		tmp = exp(Float64(Float64(exp(w) * log(l)) - w));
	else
		tmp = l ^ exp(w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.35e-14)
		tmp = exp(((exp(w) * log(l)) - w));
	else
		tmp = l ^ exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.35e-14], N[Exp[N[(N[(N[Exp[w], $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.3499999999999999e-14

   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

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. +-rgt-identity N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. unsub-neg N/A

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

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

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

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

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

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

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

      4. exp-lowering-exp.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   if -1.3499999999999999e-14 < w

   1. Initial program 98.2%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 99.0%

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

      1. *-lft-identity N/A

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

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

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

      3. exp-lowering-exp.f64 99.0

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.35 \cdot 10^{-14}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;t\_0 \leq 10^{-306}:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* (exp (- 0.0 w)) (pow l (exp w)))))
   (if (<= t_0 1e-306) 0.0 (if (<= t_0 4e+307) l (* l (* w (* w 0.5)))))))

C

double code(double w, double l) {
	double t_0 = exp((0.0 - w)) * pow(l, exp(w));
	double tmp;
	if (t_0 <= 1e-306) {
		tmp = 0.0;
	} else if (t_0 <= 4e+307) {
		tmp = l;
	} else {
		tmp = l * (w * (w * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - w)) * (l ** exp(w))
    if (t_0 <= 1d-306) then
        tmp = 0.0d0
    else if (t_0 <= 4d+307) then
        tmp = l
    else
        tmp = l * (w * (w * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.exp((0.0 - w)) * Math.pow(l, Math.exp(w));
	double tmp;
	if (t_0 <= 1e-306) {
		tmp = 0.0;
	} else if (t_0 <= 4e+307) {
		tmp = l;
	} else {
		tmp = l * (w * (w * 0.5));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.exp((0.0 - w)) * math.pow(l, math.exp(w))
	tmp = 0
	if t_0 <= 1e-306:
		tmp = 0.0
	elif t_0 <= 4e+307:
		tmp = l
	else:
		tmp = l * (w * (w * 0.5))
	return tmp

Julia

function code(w, l)
	t_0 = Float64(exp(Float64(0.0 - w)) * (l ^ exp(w)))
	tmp = 0.0
	if (t_0 <= 1e-306)
		tmp = 0.0;
	elseif (t_0 <= 4e+307)
		tmp = l;
	else
		tmp = Float64(l * Float64(w * Float64(w * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = exp((0.0 - w)) * (l ^ exp(w));
	tmp = 0.0;
	if (t_0 <= 1e-306)
		tmp = 0.0;
	elseif (t_0 <= 4e+307)
		tmp = l;
	else
		tmp = l * (w * (w * 0.5));
	end
	tmp_2 = 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, 1e-306], 0.0, If[LessEqual[t$95$0, 4e+307], l, N[(l * N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 1.00000000000000003e-306

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 97.6%

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

   if 1.00000000000000003e-306 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 3.99999999999999994e307

   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 97.0%

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

   if 3.99999999999999994e307 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 95.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 55.3

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

   5. Simplified 55.3%

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

   6. Taylor expanded in w around inf

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

      1. unpow2 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate--r- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-sub N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*l* N/A

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

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

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

      13. distribute-lft-in N/A

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

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

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac2 N/A

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

      17. associate-*r/ N/A

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

      18. *-rgt-identity N/A

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

   8. Simplified 55.3%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 55.3

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

   11. Simplified 55.3%

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

   12. Taylor expanded in w around 0

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

   14. Simplified 62.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{-306}:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- 0.0 w)) (pow l (exp w))) 1e-306)
   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))) <= 1e-306) {
		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))) <= 1e-306)
		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], 1e-306], 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))) < 1.00000000000000003e-306

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 97.6%

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

   if 1.00000000000000003e-306 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 98.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 96.6%

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

   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 86.2

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

   8. Simplified 86.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- 0.0 w)) (pow l (exp w))) 1e-306) 0.0 l))

C

double code(double w, double l) {
	double tmp;
	if ((exp((0.0 - w)) * pow(l, exp(w))) <= 1e-306) {
		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))) <= 1d-306) 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))) <= 1e-306) {
		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))) <= 1e-306:
		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))) <= 1e-306)
		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))) <= 1e-306)
		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], 1e-306], 0.0, l]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 1.00000000000000003e-306

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 97.6%

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

   if 1.00000000000000003e-306 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 98.4%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 68.4%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 17.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 1.1 \cdot 10^{-154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- 0.0 w)) (pow l (exp w))) 1.1e-154) 0.0 1.0))

C

double code(double w, double l) {
	double tmp;
	if ((exp((0.0 - w)) * pow(l, exp(w))) <= 1.1e-154) {
		tmp = 0.0;
	} else {
		tmp = 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 ((exp((0.0d0 - w)) * (l ** exp(w))) <= 1.1d-154) then
        tmp = 0.0d0
    else
        tmp = 1.0d0
    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))) <= 1.1e-154) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (math.exp((0.0 - w)) * math.pow(l, math.exp(w))) <= 1.1e-154:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(0.0 - w)) * (l ^ exp(w))) <= 1.1e-154)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp((0.0 - w)) * (l ^ exp(w))) <= 1.1e-154)
		tmp = 0.0;
	else
		tmp = 1.0;
	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], 1.1e-154], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 1.10000000000000004e-154

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 48.6%

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

   if 1.10000000000000004e-154 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 98.1%

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

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. +-rgt-identity N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. unsub-neg N/A

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

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

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

   5. Simplified 94.9%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 40.0

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

   8. Simplified 40.0%

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

   9. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 5.1%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 1.1 \cdot 10^{-154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 98.6%

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

3. Final simplification 98.6%

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

Alternative 7: 98.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.0066:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.0066)
   (pow l (fma w (fma w (fma w 0.16666666666666666 0.5) 1.0) 1.0))
   (* (fma w (fma w 0.5 -1.0) 1.0) (pow l (fma w (fma w 0.5 1.0) 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.0066) {
		tmp = pow(l, fma(w, fma(w, fma(w, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else {
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0) * pow(l, fma(w, fma(w, 0.5, 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.0066)
		tmp = l ^ fma(w, fma(w, fma(w, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(fma(w, fma(w, 0.5, -1.0), 1.0) * (l ^ fma(w, fma(w, 0.5, 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.0066], N[Power[l, N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], N[(N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[l, N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.0066

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 77.4%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 98.7

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

   8. Simplified 98.7%

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

      1. *-lft-identity N/A

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

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

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

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

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

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

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

      5. accelerator-lowering-fma.f64 98.7

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

   10. Applied egg-rr 98.7%

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

   if 0.0066 < l

   1. Initial program 97.1%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 85.8

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

   5. Simplified 85.8%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 98.8

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

   8. Simplified 98.8%

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

Alternative 8: 97.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 36000000:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (exp (- 0.0 w))
   (if (<= w 36000000.0) (* l (fma w (fma w 0.5 -1.0) 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 <= 36000000.0) {
		tmp = l * fma(w, fma(w, 0.5, -1.0), 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 <= 36000000.0)
		tmp = Float64(l * fma(w, fma(w, 0.5, -1.0), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 36000000.0], N[(l * N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in 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

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. +-rgt-identity N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. unsub-neg N/A

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

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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 98.7

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

   8. Simplified 98.7%

      \[\leadsto e^{\color{blue}{0 - w}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 98.7

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

   10. Applied egg-rr 98.7%

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

   if -0.69999999999999996 < w < 3.6e7

   1. Initial program 97.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 95.2%

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

   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 95.3

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

   8. Simplified 95.3%

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

   if 3.6e7 < 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 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 36000000:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% 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 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 96.7%

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

6. Final simplification 96.7%

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

Alternative 10: 89.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -6.6 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;w \leq 36000000:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -6.6e+101)
   (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= w 36000000.0) (* l (fma w (fma w 0.5 -1.0) 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -6.6e+101) {
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (w <= 36000000.0) {
		tmp = l * fma(w, fma(w, 0.5, -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -6.6e+101)
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (w <= 36000000.0)
		tmp = Float64(l * fma(w, fma(w, 0.5, -1.0), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -6.6e+101], N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 36000000.0], N[(l * N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -6.60000000000000022e101

   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

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. +-rgt-identity N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. unsub-neg N/A

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

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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

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

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f64 98.0

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

   11. Simplified 98.0%

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

   if -6.60000000000000022e101 < w < 3.6e7

   1. Initial program 97.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 95.3%

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

   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 86.3

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

   8. Simplified 86.3%

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

   if 3.6e7 < 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 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -6.6 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;w \leq 36000000:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.0075:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.0075)
   (* l (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.0075) {
		tmp = l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.0075)
		tmp = Float64(l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.0075], N[(l * N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.0074999999999999997

   1. Initial program 99.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 91.3

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

   8. Simplified 91.3%

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

   if 0.0074999999999999997 < w

   1. Initial program 91.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 90.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.0075:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;w \leq 0.0075:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.75e+154)
   (fma w (fma w 0.5 -1.0) 1.0)
   (if (<= w 0.0075) (* l (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.75e+154) {
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	} else if (w <= 0.0075) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.75e+154)
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	elseif (w <= 0.0075)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.75e+154], N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 0.0075], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -1.7500000000000001e154

   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

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. +-rgt-identity N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. unsub-neg N/A

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

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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

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

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 100.0

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

   11. Simplified 100.0%

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

   if -1.7500000000000001e154 < w < 0.0074999999999999997

   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 97.5%

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

   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 84.0

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

   8. Simplified 84.0%

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

   if 0.0074999999999999997 < w

   1. Initial program 91.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 90.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;w \leq 0.0075:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.0% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.0075:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.0075) (* l (- 1.0 w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.0075) {
		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 <= 0.0075d0) 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 <= 0.0075) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.0075:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.0075)
		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 <= 0.0075)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.0075], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.0074999999999999997

   1. Initial program 99.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 78.1

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

   8. Simplified 78.1%

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

   if 0.0074999999999999997 < w

   1. Initial program 91.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 90.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.0075:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 15.9% 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 98.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 16.1%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))