exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 20.3s

Alternatives: 17

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = 2.1016891902848384e+99
  2. l = -3.6881635834943054e-105

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: 99.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 69.9

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

   5. Simplified 69.9%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 75.8%

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

4. Final simplification 79.7%

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

Alternative 3: 70.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul0-lft N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. flip-- N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 69.0%

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

4. Final simplification 74.0%

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

Alternative 4: 19.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 1.12 \cdot 10^{-154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- 0.0 w)) (pow l (exp w))) 1.12e-154) 0.0 1.0))

C

double code(double w, double l) {
	double tmp;
	if ((exp((0.0 - w)) * pow(l, exp(w))) <= 1.12e-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))) <= 1.12d-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))) <= 1.12e-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))) <= 1.12e-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))) <= 1.12e-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))) <= 1.12e-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], 1.12e-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))) < 1.12e-154

   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 60.1%

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

   if 1.12e-154 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.8%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 40.6

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

   4. Applied egg-rr 40.6%

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

   5. Taylor expanded in w around 0

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

   7. Simplified 5.3%

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

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - w} \cdot {\ell}^{\left(e^{w}\right)} \leq 1.12 \cdot 10^{-154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(1) ^ Float64(0.0 - w);
	else
		tmp = Float64(Float64(1.0 - w) * Float64(l * (l ^ w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = 2.71828182845904523536 ^ (0.0 - w);
	else
		tmp = (1.0 - w) * (l * (l ^ w));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.9

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

   4. Applied egg-rr 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 98.9

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity N/A

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

      2. pow-exp N/A

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

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

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

      4. exp-1-e N/A

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

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

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

      6. --lowering--.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   if -1 < w

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 99.7

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

   8. Simplified 99.7%

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

      1. +-commutative N/A

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

      2. pow-plus N/A

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

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

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

      4. pow-lowering-pow.f64 99.8

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

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

Alternative 6: 98.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(1) ^ Float64(0.0 - w);
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(w + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = 2.71828182845904523536 ^ (0.0 - w);
	else
		tmp = (1.0 - w) * (l ^ (w + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.9

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

   4. Applied egg-rr 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 98.9

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity N/A

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

      2. pow-exp N/A

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

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

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

      4. exp-1-e N/A

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

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

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

      6. --lowering--.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   if -1 < w

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 99.7

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

   8. Simplified 99.7%

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

4. Final simplification 99.5%

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

Alternative 7: 98.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(1) ^ Float64(0.0 - w);
	else
		tmp = Float64(l * (l ^ w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = 2.71828182845904523536 ^ (0.0 - w);
	else
		tmp = l * (l ^ w);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.9

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

   4. Applied egg-rr 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 98.9

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity N/A

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

      2. pow-exp N/A

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

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

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

      4. exp-1-e N/A

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

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

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

      6. --lowering--.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   if -1 < w

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 99.7

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

   8. Simplified 99.7%

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

      1. +-commutative N/A

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

      2. pow-plus N/A

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

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

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

      4. pow-lowering-pow.f64 99.8

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in w around 0

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

   13. Simplified 99.6%

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

4. Final simplification 99.4%

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

Alternative 8: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.05:\\ \;\;\;\;{e}^{\left(0 - w\right)}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(w + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.05) (pow E (- 0.0 w)) (pow l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.05) {
		tmp = pow(((double) M_E), (0.0 - w));
	} else {
		tmp = pow(l, (w + 1.0));
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.05) {
		tmp = Math.pow(Math.E, (0.0 - w));
	} else {
		tmp = Math.pow(l, (w + 1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.05:
		tmp = math.pow(math.e, (0.0 - w))
	else:
		tmp = math.pow(l, (w + 1.0))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.05)
		tmp = exp(1) ^ Float64(0.0 - w);
	else
		tmp = l ^ Float64(w + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.05)
		tmp = 2.71828182845904523536 ^ (0.0 - w);
	else
		tmp = l ^ (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.05], N[Power[E, N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.05000000000000004

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.9

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

   4. Applied egg-rr 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 98.9

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity N/A

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

      2. pow-exp N/A

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

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

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

      4. exp-1-e N/A

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

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

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

      6. --lowering--.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   if -1.05000000000000004 < w

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in w around 0

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

      1. +-lowering-+.f64 99.7

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

   8. Simplified 99.7%

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

   9. Taylor expanded in w around 0

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

   11. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.05:\\ \;\;\;\;{e}^{\left(0 - w\right)}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(w + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;{e}^{\left(0 - w\right)}\\ \mathbf{elif}\;w \leq 0.145:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68) (pow E (- 0.0 w)) (if (<= w 0.145) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = pow(((double) M_E), (0.0 - w));
	} else if (w <= 0.145) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = Math.pow(Math.E, (0.0 - w));
	} else if (w <= 0.145) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = math.pow(math.e, (0.0 - w))
	elif w <= 0.145:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = exp(1) ^ Float64(0.0 - w);
	elseif (w <= 0.145)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = 2.71828182845904523536 ^ (0.0 - w);
	elseif (w <= 0.145)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[Power[E, N[(0.0 - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.145], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.680000000000000049

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.9

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

   4. Applied egg-rr 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 98.9

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity N/A

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

      2. pow-exp N/A

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

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

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

      4. exp-1-e N/A

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

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

         \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(0 - w\right)} \]

      6. --lowering--.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   if -0.680000000000000049 < w < 0.14499999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 99.2%

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

   if 0.14499999999999999 < 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. Add Preprocessing

Alternative 10: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.205:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7) (exp (- 0.0 w)) (if (<= w 0.205) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp((0.0 - w));
	} else if (w <= 0.205) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.7d0)) then
        tmp = exp((0.0d0 - w))
    else if (w <= 0.205d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = Math.exp((0.0 - w));
	} else if (w <= 0.205) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = math.exp((0.0 - w))
	elif w <= 0.205:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(0.0 - w));
	elseif (w <= 0.205)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.7)
		tmp = exp((0.0 - w));
	elseif (w <= 0.205)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.205], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.9

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

   4. Applied egg-rr 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 98.9

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

   6. Applied egg-rr 98.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   if -0.69999999999999996 < w < 0.204999999999999988

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 99.2%

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

   if 0.204999999999999988 < 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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.205:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right)\\ \mathbf{if}\;w \leq -1 \cdot 10^{+103}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq -0.68:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0 \cdot \mathsf{fma}\left(w, w, 0\right), -1\right)}{\mathsf{fma}\left(w, t\_0, -1\right)}\\ \mathbf{elif}\;w \leq 0.116:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma w (fma w -0.16666666666666666 0.5) -1.0)))
   (if (<= w -1e+103)
     (* -0.16666666666666666 (* w (* w w)))
     (if (<= w -0.68)
       (/ (fma t_0 (* t_0 (fma w w 0.0)) -1.0) (fma w t_0 -1.0))
       (if (<= w 0.116) l 0.0)))))

C

double code(double w, double l) {
	double t_0 = fma(w, fma(w, -0.16666666666666666, 0.5), -1.0);
	double tmp;
	if (w <= -1e+103) {
		tmp = -0.16666666666666666 * (w * (w * w));
	} else if (w <= -0.68) {
		tmp = fma(t_0, (t_0 * fma(w, w, 0.0)), -1.0) / fma(w, t_0, -1.0);
	} else if (w <= 0.116) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(w, fma(w, -0.16666666666666666, 0.5), -1.0)
	tmp = 0.0
	if (w <= -1e+103)
		tmp = Float64(-0.16666666666666666 * Float64(w * Float64(w * w)));
	elseif (w <= -0.68)
		tmp = Float64(fma(t_0, Float64(t_0 * fma(w, w, 0.0)), -1.0) / fma(w, t_0, -1.0));
	elseif (w <= 0.116)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[w, -1e+103], N[(-0.16666666666666666 * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -0.68], N[(N[(t$95$0 * N[(t$95$0 * N[(w * w + 0.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(w * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.116], l, 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if w < -1e103

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. 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 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {w}^{3}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {w}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 100.0

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

   10. Simplified 100.0%

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

   if -1e103 < w < -0.680000000000000049

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 97.0

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. 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 5.4

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

   7. Simplified 5.4%

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

      1. flip-+ N/A

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

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

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

   9. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right) \cdot \mathsf{fma}\left(w, w, 0\right), -1\right)}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), -1\right)}} \]

   if -0.680000000000000049 < w < 0.116000000000000006

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 99.2%

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

   if 0.116000000000000006 < 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 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 92.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right)\\ \mathbf{if}\;w \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\left(w \cdot w\right) \cdot 0.5\\ \mathbf{elif}\;w \leq -7.1 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \mathsf{fma}\left(t\_0, t\_0, -1\right), \frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), 1\right)}, 1\right)\\ \mathbf{elif}\;w \leq 0.32:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (fma w -0.16666666666666666 0.5))))
   (if (<= w -4e+154)
     (* (* w w) 0.5)
     (if (<= w -7.1e+56)
       (fma
        (* w (fma t_0 t_0 -1.0))
        (/ 1.0 (fma w (fma w -0.16666666666666666 0.5) 1.0))
        1.0)
       (if (<= w 0.32) (* l (- 1.0 w)) 0.0)))))

C

double code(double w, double l) {
	double t_0 = w * fma(w, -0.16666666666666666, 0.5);
	double tmp;
	if (w <= -4e+154) {
		tmp = (w * w) * 0.5;
	} else if (w <= -7.1e+56) {
		tmp = fma((w * fma(t_0, t_0, -1.0)), (1.0 / fma(w, fma(w, -0.16666666666666666, 0.5), 1.0)), 1.0);
	} else if (w <= 0.32) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = Float64(w * fma(w, -0.16666666666666666, 0.5))
	tmp = 0.0
	if (w <= -4e+154)
		tmp = Float64(Float64(w * w) * 0.5);
	elseif (w <= -7.1e+56)
		tmp = fma(Float64(w * fma(t_0, t_0, -1.0)), Float64(1.0 / fma(w, fma(w, -0.16666666666666666, 0.5), 1.0)), 1.0);
	elseif (w <= 0.32)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4e+154], N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[w, -7.1e+56], N[(N[(w * N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 0.32], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if w < -4.00000000000000015e154

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0

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

   10. Simplified 100.0%

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

   if -4.00000000000000015e154 < w < -7.1e56

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. 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 59.2

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

   7. Simplified 59.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   9. Applied egg-rr 91.7%

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

   if -7.1e56 < w < 0.320000000000000007

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 90.9%

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

   if 0.320000000000000007 < 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 4 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\left(w \cdot w\right) \cdot 0.5\\ \mathbf{elif}\;w \leq -7.1 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \mathsf{fma}\left(w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), 1\right)}, 1\right)\\ \mathbf{elif}\;w \leq 0.32:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(w, w, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(w, w, 0\right), 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(w, -0.16666666666666666, -0.5\right)}, 1 - w\right)\\ \mathbf{elif}\;w \leq 0.13:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell \cdot \left(1 - \mathsf{fma}\left(w, w, 0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.1e+71)
   (fma
    (* (fma w w 0.0) (fma (fma w w 0.0) 0.027777777777777776 -0.25))
    (/ 1.0 (fma w -0.16666666666666666 -0.5))
    (- 1.0 w))
   (if (<= w 0.13) (/ 1.0 (/ (+ w 1.0) (* l (- 1.0 (fma w w 0.0))))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.1e+71) {
		tmp = fma((fma(w, w, 0.0) * fma(fma(w, w, 0.0), 0.027777777777777776, -0.25)), (1.0 / fma(w, -0.16666666666666666, -0.5)), (1.0 - w));
	} else if (w <= 0.13) {
		tmp = 1.0 / ((w + 1.0) / (l * (1.0 - fma(w, w, 0.0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.1e+71)
		tmp = fma(Float64(fma(w, w, 0.0) * fma(fma(w, w, 0.0), 0.027777777777777776, -0.25)), Float64(1.0 / fma(w, -0.16666666666666666, -0.5)), Float64(1.0 - w));
	elseif (w <= 0.13)
		tmp = Float64(1.0 / Float64(Float64(w + 1.0) / Float64(l * Float64(1.0 - fma(w, w, 0.0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.1e+71], N[(N[(N[(w * w + 0.0), $MachinePrecision] * N[(N[(w * w + 0.0), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(w * -0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 - w), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.13], N[(1.0 / N[(N[(w + 1.0), $MachinePrecision] / N[(l * N[(1.0 - N[(w * w + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -1.09999999999999997e71

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. 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 88.6

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

   7. Simplified 88.6%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. neg-mul-1 N/A

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

      9. sub0-neg N/A

         \[\leadsto \left(\left(w \cdot w\right) \cdot \left(\left(w \cdot \frac{-1}{6}\right) \cdot \left(w \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{w \cdot \frac{-1}{6} - \frac{1}{2}} + \left(\color{blue}{\left(0 - w\right)} + 1\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(w \cdot w\right) \cdot \left(\left(w \cdot \frac{-1}{6}\right) \cdot \left(w \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{w \cdot \frac{-1}{6} - \frac{1}{2}} + \color{blue}{\left(1 + \left(0 - w\right)\right)} \]

      11. sub0-neg N/A

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

      12. sub-neg N/A

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

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

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

   9. Applied egg-rr 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(w, w, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(w, w, 0\right), 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(w, -0.16666666666666666, -0.5\right)}, 1 - w\right)} \]

   if -1.09999999999999997e71 < w < 0.13

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 88.8

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

   5. Simplified 88.8%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. +-lft-identity N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-out N/A

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

      16. mul0-lft N/A

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

      17. accelerator-lowering-fma.f64 88.6

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

   7. Applied egg-rr 88.6%

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

   8. Taylor expanded in w around 0

      \[\leadsto \frac{1}{\frac{1 + w}{\color{blue}{\ell} \cdot \left(1 - \mathsf{fma}\left(w, w, 0\right)\right)}} \]
   9. Step-by-step derivation

   10. Simplified 89.1%

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

   if 0.13 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      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 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(w, w, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(w, w, 0\right), 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(w, -0.16666666666666666, -0.5\right)}, 1 - w\right)\\ \mathbf{elif}\;w \leq 0.13:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell \cdot \left(1 - \mathsf{fma}\left(w, w, 0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -5.6 \cdot 10^{+92}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq 0.225:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell \cdot \left(1 - \mathsf{fma}\left(w, w, 0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -5.6e+92)
   (* -0.16666666666666666 (* w (* w w)))
   (if (<= w 0.225) (/ 1.0 (/ (+ w 1.0) (* l (- 1.0 (fma w w 0.0))))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -5.6e+92) {
		tmp = -0.16666666666666666 * (w * (w * w));
	} else if (w <= 0.225) {
		tmp = 1.0 / ((w + 1.0) / (l * (1.0 - fma(w, w, 0.0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -5.6e+92)
		tmp = Float64(-0.16666666666666666 * Float64(w * Float64(w * w)));
	elseif (w <= 0.225)
		tmp = Float64(1.0 / Float64(Float64(w + 1.0) / Float64(l * Float64(1.0 - fma(w, w, 0.0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -5.6e+92], N[(-0.16666666666666666 * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.225], N[(1.0 / N[(N[(w + 1.0), $MachinePrecision] / N[(l * N[(1.0 - N[(w * w + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -5.60000000000000001e92

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. 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 97.9

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

   7. Simplified 97.9%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {w}^{3}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {w}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.9

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

   10. Simplified 97.9%

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

   if -5.60000000000000001e92 < w < 0.225000000000000006

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 86.3

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

   5. Simplified 86.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. +-lft-identity N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-out N/A

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

      16. mul0-lft N/A

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

      17. accelerator-lowering-fma.f64 86.1

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in w around 0

      \[\leadsto \frac{1}{\frac{1 + w}{\color{blue}{\ell} \cdot \left(1 - \mathsf{fma}\left(w, w, 0\right)\right)}} \]
   9. Step-by-step derivation

   10. Simplified 87.1%

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

   if 0.225000000000000006 < 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 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -5.6 \cdot 10^{+92}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq 0.225:\\ \;\;\;\;\frac{1}{\frac{w + 1}{\ell \cdot \left(1 - \mathsf{fma}\left(w, w, 0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.1% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -5.9 \cdot 10^{+56}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq 0.085:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -5.9e+56)
   (* -0.16666666666666666 (* w (* w w)))
   (if (<= w 0.085) (* l (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -5.9e+56) {
		tmp = -0.16666666666666666 * (w * (w * w));
	} else if (w <= 0.085) {
		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 <= (-5.9d+56)) then
        tmp = (-0.16666666666666666d0) * (w * (w * w))
    else if (w <= 0.085d0) 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 <= -5.9e+56) {
		tmp = -0.16666666666666666 * (w * (w * w));
	} else if (w <= 0.085) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -5.9e+56:
		tmp = -0.16666666666666666 * (w * (w * w))
	elif w <= 0.085:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -5.9e+56)
		tmp = Float64(-0.16666666666666666 * Float64(w * Float64(w * w)));
	elseif (w <= 0.085)
		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 <= -5.9e+56)
		tmp = -0.16666666666666666 * (w * (w * w));
	elseif (w <= 0.085)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -5.9e+56], N[(-0.16666666666666666 * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.085], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -5.9000000000000001e56

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)} \]
   6. 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 82.3

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

   7. Simplified 82.3%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {w}^{3}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {w}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 82.3

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

   10. Simplified 82.3%

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

   if -5.9000000000000001e56 < w < 0.0850000000000000061

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 90.9%

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

   if 0.0850000000000000061 < 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 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -5.9 \cdot 10^{+56}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq 0.085:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.45 \cdot 10^{+144}:\\ \;\;\;\;\left(w \cdot w\right) \cdot 0.5\\ \mathbf{elif}\;w \leq 0.102:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.45e+144) (* (* w w) 0.5) (if (<= w 0.102) (* l (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.45e+144) {
		tmp = (w * w) * 0.5;
	} else if (w <= 0.102) {
		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 <= (-2.45d+144)) then
        tmp = (w * w) * 0.5d0
    else if (w <= 0.102d0) 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 <= -2.45e+144) {
		tmp = (w * w) * 0.5;
	} else if (w <= 0.102) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.45e+144:
		tmp = (w * w) * 0.5
	elif w <= 0.102:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.45e+144)
		tmp = Float64(Float64(w * w) * 0.5);
	elseif (w <= 0.102)
		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 <= -2.45e+144)
		tmp = (w * w) * 0.5;
	elseif (w <= 0.102)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.45e+144], N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[w, 0.102], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -2.45e144

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. pow-prod-up N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)} \]
   6. 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 89.5

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

   7. Simplified 89.5%

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

   8. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {w}^{2}} \]

      2. unpow2 N/A

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

      3. *-lowering-*.f64 89.5

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

   10. Simplified 89.5%

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

   if -2.45e144 < w < 0.101999999999999993

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 81.8

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

   5. Simplified 81.8%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 82.3%

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

   if 0.101999999999999993 < 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 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.45 \cdot 10^{+144}:\\ \;\;\;\;\left(w \cdot w\right) \cdot 0.5\\ \mathbf{elif}\;w \leq 0.102:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 17.2% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

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

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 18.6%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))