exp-w (used to crash)

Percentage Accurate: 99.4% → 99.2%

Time: 20.9s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = 1.73448873005108e+26
  2. l = -2.35992677628926e-122

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.55 \cdot 10^{-16}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(w + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.55e-16)
   (exp (fma (log l) (exp w) (- w)))
   (* 1.0 (pow l (+ w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.55e-16) {
		tmp = exp(fma(log(l), exp(w), -w));
	} else {
		tmp = 1.0 * pow(l, (w + 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.55e-16)
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	else
		tmp = Float64(1.0 * (l ^ Float64(w + 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.55e-16], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision], N[(1.0 * N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -2.55e-16

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -2.55e-16 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.55 \cdot 10^{-16}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(w + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if ((exp(-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(-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(-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(-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(-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(-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[(-w)], $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

   4. Applied rewrites 100.0%

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

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

   1. Initial program 99.2%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 94.3

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

   8. Applied rewrites 94.3%

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

   10. Applied rewrites 94.2%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 64.6%

       \[\leadsto \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 18.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{-155}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 5e-155) 0.0 1.0))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-155) {
		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(-w) * (l ** exp(w))) <= 5d-155) 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(-w) * Math.pow(l, Math.exp(w))) <= 5e-155) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (math.exp(-w) * math.pow(l, math.exp(w))) <= 5e-155:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 5e-155)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp(-w) * (l ^ exp(w))) <= 5e-155)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-155], 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))) < 4.9999999999999999e-155

   1. Initial program 99.7%

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

   4. Applied rewrites 57.4%

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

   if 4.9999999999999999e-155 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 4.8%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

Alternative 5: 98.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;{\ell}^{\left(w + 1\right)} \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 7.2e-5)
   (* (pow l (+ w 1.0)) (- 1.0 w))
   (* (fma w (fma w 0.5 -1.0) 1.0) (pow l (fma w (fma w 0.5 1.0) 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 7.2e-5) {
		tmp = pow(l, (w + 1.0)) * (1.0 - w);
	} else {
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0) * pow(l, fma(w, fma(w, 0.5, 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 7.2e-5)
		tmp = Float64((l ^ Float64(w + 1.0)) * Float64(1.0 - w));
	else
		tmp = Float64(fma(w, fma(w, 0.5, -1.0), 1.0) * (l ^ fma(w, fma(w, 0.5, 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 7.2e-5], N[(N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[l, N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.20000000000000018e-5

   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. lower--.f64 72.0

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

   5. Applied rewrites 72.0%

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

   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. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 7.20000000000000018e-5 < l

   1. Initial program 98.7%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;{\ell}^{\left(w + 1\right)} \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[(-w)], $MachinePrecision], N[(1.0 * 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. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in w around inf

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

   11. Applied rewrites 99.0%

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

   if -1 < w

   1. Initial program 99.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   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. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 99.0%

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

4. Final simplification 99.0%

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

Alternative 7: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7) (exp (- w)) (if (<= w 135000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp(-w);
	} else if (w <= 135000.0) {
		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(-w)
    else if (w <= 135000.0d0) 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(-w);
	} else if (w <= 135000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = math.exp(-w)
	elif w <= 135000.0:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(-w));
	elseif (w <= 135000.0)
		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(-w);
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 135000.0], 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. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in w around inf

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

   11. Applied rewrites 99.0%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 8: 93.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right)\\ t_1 := w \cdot t\_0\\ \mathbf{if}\;w \leq -1 \cdot 10^{+103}:\\ \;\;\;\;w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq -0.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(w, t\_0, -1\right)}\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma w (fma w -0.16666666666666666 0.5) -1.0)) (t_1 (* w t_0)))
   (if (<= w -1e+103)
     (* w (* -0.16666666666666666 (* w w)))
     (if (<= w -0.7)
       (/ (fma t_1 t_1 -1.0) (fma w t_0 -1.0))
       (if (<= w 135000.0) l 0.0)))))

C

double code(double w, double l) {
	double t_0 = fma(w, fma(w, -0.16666666666666666, 0.5), -1.0);
	double t_1 = w * t_0;
	double tmp;
	if (w <= -1e+103) {
		tmp = w * (-0.16666666666666666 * (w * w));
	} else if (w <= -0.7) {
		tmp = fma(t_1, t_1, -1.0) / fma(w, t_0, -1.0);
	} else if (w <= 135000.0) {
		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)
	t_1 = Float64(w * t_0)
	tmp = 0.0
	if (w <= -1e+103)
		tmp = Float64(w * Float64(-0.16666666666666666 * Float64(w * w)));
	elseif (w <= -0.7)
		tmp = Float64(fma(t_1, t_1, -1.0) / fma(w, t_0, -1.0));
	elseif (w <= 135000.0)
		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]}, Block[{t$95$1 = N[(w * t$95$0), $MachinePrecision]}, If[LessEqual[w, -1e+103], N[(w * N[(-0.16666666666666666 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -0.7], N[(N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision] / N[(w * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 135000.0], 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 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. lower-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. lower-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. lower-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. Applied rewrites 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

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

   10. Applied rewrites 100.0%

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

   if -1e103 < 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 97.5

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

   4. Applied rewrites 97.5%

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

   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. lower-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. lower-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. lower-fma.f64 5.5

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

   7. Applied rewrites 5.5%

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

   9. Applied rewrites 43.2%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 9: 90.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right)\\ \mathbf{if}\;w \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;w \leq -0.7:\\ \;\;\;\;\mathsf{fma}\left(w, \frac{\mathsf{fma}\left(t\_0, t\_0, -1\right)}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), 1\right)}, 1\right)\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (fma w -0.16666666666666666 0.5))))
   (if (<= w -1e+155)
     (fma w (fma w 0.5 -1.0) 1.0)
     (if (<= w -0.7)
       (fma
        w
        (/ (fma t_0 t_0 -1.0) (fma w (fma w -0.16666666666666666 0.5) 1.0))
        1.0)
       (if (<= w 135000.0) l 0.0)))))

C

double code(double w, double l) {
	double t_0 = w * fma(w, -0.16666666666666666, 0.5);
	double tmp;
	if (w <= -1e+155) {
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	} else if (w <= -0.7) {
		tmp = fma(w, (fma(t_0, t_0, -1.0) / fma(w, fma(w, -0.16666666666666666, 0.5), 1.0)), 1.0);
	} else if (w <= 135000.0) {
		tmp = l;
	} 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 <= -1e+155)
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	elseif (w <= -0.7)
		tmp = fma(w, Float64(fma(t_0, t_0, -1.0) / fma(w, fma(w, -0.16666666666666666, 0.5), 1.0)), 1.0);
	elseif (w <= 135000.0)
		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]), $MachinePrecision]}, If[LessEqual[w, -1e+155], N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, -0.7], N[(w * N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 135000.0], l, 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if w < -1.00000000000000001e155

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 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. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-lft-in N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.00000000000000001e155 < 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.2

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

   4. Applied rewrites 98.2%

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

   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. lower-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. lower-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. lower-fma.f64 29.7

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

   7. Applied rewrites 29.7%

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

   9. Applied rewrites 46.8%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 10: 88.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= w 135000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (w <= 135000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 135000.0], 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   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. lower-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. lower-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. lower-fma.f64 60.7

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

   7. Applied rewrites 60.7%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 11: 88.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, w \cdot -0.16666666666666666, -1\right), 1\right)\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (fma w (fma w (* w -0.16666666666666666) -1.0) 1.0)
   (if (<= w 135000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = fma(w, fma(w, (w * -0.16666666666666666), -1.0), 1.0);
	} else if (w <= 135000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = fma(w, fma(w, Float64(w * -0.16666666666666666), -1.0), 1.0);
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[(w * N[(w * N[(w * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 135000.0], 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   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. lower-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. lower-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. lower-fma.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 60.7%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 12: 88.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;\mathsf{fma}\left(w, -0.16666666666666666 \cdot \left(w \cdot w\right), 1\right)\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (fma w (* -0.16666666666666666 (* w w)) 1.0)
   (if (<= w 135000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = fma(w, (-0.16666666666666666 * (w * w)), 1.0);
	} else if (w <= 135000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = fma(w, Float64(-0.16666666666666666 * Float64(w * w)), 1.0);
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[(w * N[(-0.16666666666666666 * N[(w * w), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 135000.0], 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   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. lower-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. lower-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. lower-fma.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 60.7%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 13: 88.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.72:\\ \;\;\;\;w \cdot \left(-0.16666666666666666 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.72)
   (* w (* -0.16666666666666666 (* w w)))
   (if (<= w 135000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.72) {
		tmp = w * (-0.16666666666666666 * (w * w));
	} else if (w <= 135000.0) {
		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 <= (-0.72d0)) then
        tmp = w * ((-0.16666666666666666d0) * (w * w))
    else if (w <= 135000.0d0) 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 <= -0.72) {
		tmp = w * (-0.16666666666666666 * (w * w));
	} else if (w <= 135000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.72:
		tmp = w * (-0.16666666666666666 * (w * w))
	elif w <= 135000.0:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.72)
		tmp = Float64(w * Float64(-0.16666666666666666 * Float64(w * w)));
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.72)
		tmp = w * (-0.16666666666666666 * (w * w));
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.72], N[(w * N[(-0.16666666666666666 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 135000.0], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.71999999999999997

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   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. lower-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. lower-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. lower-fma.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 60.7%

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

   if -0.71999999999999997 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 14: 84.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;w \leq 135000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7) (fma w (fma w 0.5 -1.0) 1.0) (if (<= w 135000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	} else if (w <= 135000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	elseif (w <= 135000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 135000.0], 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. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   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. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-lft-in N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 47.1

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

   7. Applied rewrites 47.1%

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

   if -0.69999999999999996 < w < 135000

   1. Initial program 98.8%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 91.2

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

   8. Applied rewrites 91.2%

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 97.5%

       \[\leadsto \ell \]

   if 135000 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 15: 16.4% 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 99.3%

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

4. Applied rewrites 16.5%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))