exp-w (used to crash)

Percentage Accurate: 99.4% → 99.6%

Time: 17.7s

Alternatives: 16

Speedup: 1.3×

• could not determine a ground truth

  1. w = 3.4510980693297704e+213
  2. l = 4.329106375005348e+222

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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.05 \cdot 10^{-9}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.05e-9)
   (exp (fma (log l) (exp w) (- w)))
   (/
    (pow l (exp w))
    (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.05e-9) {
		tmp = exp(fma(log(l), exp(w), -w));
	} else {
		tmp = pow(l, exp(w)) / fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.05e-9)
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	else
		tmp = Float64((l ^ exp(w)) / fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.05e-9], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -2.0500000000000002e-9

   1. Initial program 99.7%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. sub-neg N/A

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

      11. +-rgt-identity N/A

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

      12. div-exp N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if -2.0500000000000002e-9 < w

   1. Initial program 99.1%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. sub-neg N/A

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

      11. +-rgt-identity N/A

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

      12. div-exp N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 99.5%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), \color{blue}{w}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in w around inf

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

   1. *-commutative N/A

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

   2. exp-to-pow N/A

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

   3. remove-double-neg N/A

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

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

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

   5. log-rec N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. +-rgt-identity N/A

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

   9. exp-sum N/A

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

   10. sub-neg N/A

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

   11. +-rgt-identity N/A

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

   12. div-exp N/A

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

   13. lower-/.f64 N/A

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

5. Applied rewrites 99.3%

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

Alternative 3: 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 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.56:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.56)
   (exp (- w))
   (/
    (pow l (exp w))
    (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.56) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, exp(w)) / fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.56)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.56], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.5600000000000001

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1.5600000000000001 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. sub-neg N/A

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

      11. +-rgt-identity N/A

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

      12. div-exp N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 99.1%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), \color{blue}{w}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.0) (exp (- w)) (/ (pow l (exp w)) (fma (fma 0.5 w 1.0) w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, exp(w)) / fma(fma(0.5, w, 1.0), w, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / fma(fma(0.5, w, 1.0), w, 1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -4 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. sub-neg N/A

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

      11. +-rgt-identity N/A

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

      12. div-exp N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 99.0%

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

Alternative 6: 99.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. sub-neg N/A

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

      11. +-rgt-identity N/A

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

      12. div-exp N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 99.0%

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

Alternative 7: 98.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\\ \mathbf{if}\;w \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 w 1.0) w 1.0)))
   (if (<= w -5.2e+14) (exp (- w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = fma(fma(0.5, w, 1.0), w, 1.0);
	double tmp;
	if (w <= -5.2e+14) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, t_0) / t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(fma(0.5, w, 1.0), w, 1.0)
	tmp = 0.0
	if (w <= -5.2e+14)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[w, -5.2e+14], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -5.2e14

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -5.2e14 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

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

      9. exp-sum N/A

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

      10. sub-neg N/A

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

      11. +-rgt-identity N/A

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

      12. div-exp N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 98.8%

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

Alternative 8: 98.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.7

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

   8. Applied rewrites 98.7%

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

   if 1 < l

   1. Initial program 98.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.1

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

   8. Applied rewrites 99.1%

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

4. Final simplification 98.8%

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

Alternative 9: 97.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 45000000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 45000000.0)))
   (exp (- w))
   (* 1.0 (pow l 1.0))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 45000000.0)) {
		tmp = exp(-w);
	} else {
		tmp = 1.0 * pow(l, 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 <= (-0.68d0)) .or. (.not. (w <= 45000000.0d0))) then
        tmp = exp(-w)
    else
        tmp = 1.0d0 * (l ** 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 45000000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = 1.0 * Math.pow(l, 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 45000000.0):
		tmp = math.exp(-w)
	else:
		tmp = 1.0 * math.pow(l, 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 45000000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(1.0 * (l ^ 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 45000000.0)))
		tmp = exp(-w);
	else
		tmp = 1.0 * (l ^ 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 45000000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(1.0 * N[Power[l, 1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 4.5e7 < w

   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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -0.680000000000000049 < w < 4.5e7

   1. Initial program 98.7%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 94.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 45000000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = math.exp(-w)
	else:
		tmp = 1.0 * math.pow(l, (1.0 + w))
	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(1.0 + w)));
	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 ^ (1.0 + w));
	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[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 97.7

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

   8. Applied rewrites 97.7%

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

4. Final simplification 98.4%

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

Alternative 11: 45.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 44.2

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

4. Applied rewrites 44.2%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 44.2

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

6. Applied rewrites 44.2%

   \[\leadsto \color{blue}{e^{-w}} \]
7. Add Preprocessing

Alternative 12: 22.6% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (fma (fma (fma -0.16666666666666666 w 0.5) w -1.0) w 1.0))

C

double code(double w, double l) {
	return fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
}

Julia

function code(w, l)
	return fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0)
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 44.2

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

4. Applied rewrites 44.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 22.4

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

7. Applied rewrites 22.4%

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

Alternative 13: 18.2% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 \cdot w, w, 1\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (fma (* 0.5 w) w 1.0))

C

double code(double w, double l) {
	return fma((0.5 * w), w, 1.0);
}

Julia

function code(w, l)
	return fma(Float64(0.5 * w), w, 1.0)
end

Wolfram

code[w_, l_] := N[(N[(0.5 * w), $MachinePrecision] * w + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 44.2

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

4. Applied rewrites 44.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 19.0

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

7. Applied rewrites 19.0%

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

8. Taylor expanded in w around inf

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

10. Applied rewrites 19.0%

    \[\leadsto \mathsf{fma}\left(0.5 \cdot w, w, 1\right) \]
11. Add Preprocessing

Alternative 14: 17.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \left(w \cdot w\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (* w w) 0.5))

C

double code(double w, double l) {
	return (w * w) * 0.5;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (w * w) * 0.5d0
end function

Java

public static double code(double w, double l) {
	return (w * w) * 0.5;
}

Python

def code(w, l):
	return (w * w) * 0.5

Julia

function code(w, l)
	return Float64(Float64(w * w) * 0.5)
end

MATLAB

function tmp = code(w, l)
	tmp = (w * w) * 0.5;
end

Wolfram

code[w_, l_] := N[(N[(w * w), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 44.2

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

4. Applied rewrites 44.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 19.0

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

7. Applied rewrites 19.0%

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

8. Taylor expanded in w around inf

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

10. Applied rewrites 18.4%

    \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{0.5} \]
11. Add Preprocessing

Alternative 15: 4.9% accurate, 77.3× speedup

Math

\[\begin{array}{l} \\ 1 - w \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (- 1.0 w))

C

double code(double w, double l) {
	return 1.0 - w;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 1.0d0 - w
end function

Java

public static double code(double w, double l) {
	return 1.0 - w;
}

Python

def code(w, l):
	return 1.0 - w

Julia

function code(w, l)
	return Float64(1.0 - w)
end

MATLAB

function tmp = code(w, l)
	tmp = 1.0 - w;
end

Wolfram

code[w_, l_] := N[(1.0 - w), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 44.2

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

4. Applied rewrites 44.2%

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

5. Taylor expanded in w around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{1 - w} \]

   3. lower--.f64 4.9

      \[\leadsto \color{blue}{1 - w} \]

7. Applied rewrites 4.9%

   \[\leadsto \color{blue}{1 - w} \]
8. Add Preprocessing

Alternative 16: 4.4% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 1.0)

C

double code(double w, double l) {
	return 1.0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 1.0d0
end function

Java

public static double code(double w, double l) {
	return 1.0;
}

Python

def code(w, l):
	return 1.0

Julia

function code(w, l)
	return 1.0
end

MATLAB

function tmp = code(w, l)
	tmp = 1.0;
end

Wolfram

code[w_, l_] := 1.0

Derivation

1. Initial program 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 44.2

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

4. Applied rewrites 44.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 19.0

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

7. Applied rewrites 19.0%

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

8. Taylor expanded in w around 0

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

10. Applied rewrites 4.4%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024309 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))