exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 19.0s

Alternatives: 18

Speedup: 1.0×

• could not determine a ground truth

  1. w = 3.853926095879488e+282
  2. l = 1.5799993362599553e+50

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

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 99.5%

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

3. Taylor expanded in l around -inf

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. exp-diff N/A

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

   7. rem-exp-log N/A

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

   8. lower-/.f64 N/A

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

   9. rem-exp-log N/A

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

   10. lower-/.f64 N/A

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

   11. lower-exp.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-exp.f64 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 99.3

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

7. Applied rewrites 99.5%

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

Alternative 2: 37.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 2e-156)
   0.0
   (fma (fma (fma -0.16666666666666666 w 0.5) w -1.0) w 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 2e-156) {
		tmp = 0.0;
	} else {
		tmp = 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 (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 2e-156)
		tmp = 0.0;
	else
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-156], 0.0, N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w + -1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 2.00000000000000008e-156

   1. Initial program 99.4%

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

   4. Applied rewrites 49.6%

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

   if 2.00000000000000008e-156 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 37.7

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

   4. Applied rewrites 37.7%

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

   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 27.8

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

   7. Applied rewrites 27.8%

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

Alternative 3: 33.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 2e-156)
   0.0
   (fma (fma 0.5 w -1.0) w 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 2e-156) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(0.5, w, -1.0), w, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 2e-156)
		tmp = 0.0;
	else
		tmp = fma(fma(0.5, w, -1.0), w, 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-156], 0.0, N[(N[(0.5 * w + -1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 2.00000000000000008e-156

   1. Initial program 99.4%

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

   4. Applied rewrites 49.6%

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

   if 2.00000000000000008e-156 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 37.7

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

   4. Applied rewrites 37.7%

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

   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 23.3

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

   7. Applied rewrites 23.3%

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

Alternative 4: 19.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 2e-156) 0.0 (- 1.0 w)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 2e-156) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - w;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((exp(-w) * (l ** exp(w))) <= 2d-156) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - w
    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))) <= 2e-156) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (math.exp(-w) * math.pow(l, math.exp(w))) <= 2e-156:
		tmp = 0.0
	else:
		tmp = 1.0 - w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 2e-156)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp(-w) * (l ^ exp(w))) <= 2e-156)
		tmp = 0.0;
	else
		tmp = 1.0 - w;
	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], 2e-156], 0.0, N[(1.0 - w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 2.00000000000000008e-156

   1. Initial program 99.4%

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

   4. Applied rewrites 49.6%

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

   if 2.00000000000000008e-156 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 37.7

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

   4. Applied rewrites 37.7%

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

   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. sub-neg N/A

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

      3. lower--.f64 5.7

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

   7. Applied rewrites 5.7%

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

Alternative 5: 18.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((exp(-w) * (l ** exp(w))) <= 1.1d-154) then
        tmp = 0.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((Math.exp(-w) * Math.pow(l, Math.exp(w))) <= 1.1e-154) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.1e-154], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   4. Applied rewrites 49.6%

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

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

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 37.7

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 5.0%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

Alternative 7: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. exp-diff N/A

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

      7. rem-exp-log N/A

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

      8. lower-/.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 99.1

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.3

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

   10. Applied rewrites 99.3%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 99.3

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

   12. Applied rewrites 99.3%

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

Alternative 8: 99.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (/ 1.0 (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 <= -1.6) {
		tmp = 1.0 / 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 <= -1.6)
		tmp = Float64(1.0 / exp(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, -1.6], N[(1.0 / N[Exp[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 < -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

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. exp-diff N/A

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

      7. rem-exp-log N/A

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

      8. lower-/.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 99.1

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.3

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

   10. Applied rewrites 99.3%

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

Alternative 9: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)\\ \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (if (<= w -1.6) (/ 1.0 (exp w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0);
	double tmp;
	if (w <= -1.6) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, t_0) / t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)
	tmp = 0.0
	if (w <= -1.6)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[w, -1.6], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $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

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. exp-diff N/A

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

      7. rem-exp-log N/A

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

      8. lower-/.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 99.1

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.3

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 99.1%

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

Alternative 10: 99.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3)
   (/ 1.0 (exp w))
   (/
    (pow l (fma (fma 0.5 w 1.0) w 1.0))
    (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 <= -1.3) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, fma(fma(0.5, w, 1.0), w, 1.0)) / 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 <= -1.3)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = Float64((l ^ fma(fma(0.5, w, 1.0), w, 1.0)) / 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, -1.3], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $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 < -1.30000000000000004

   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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1.30000000000000004 < w

   1. Initial program 99.4%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. exp-diff N/A

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

      7. rem-exp-log N/A

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

      8. lower-/.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 99.1

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.3

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in w around 0

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

   13. Applied rewrites 98.8%

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

Alternative 11: 98.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (/ 1.0 (exp w))
   (*
    (- 1.0 w)
    (pow l (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 <= -1.6) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = (1.0 - w) * pow(l, 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 <= -1.6)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ 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, -1.6], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $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

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 12: 98.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3)
   (/ 1.0 (exp w))
   (* (- 1.0 w) (pow l (fma (fma 0.5 w 1.0) w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = (1.0 - w) * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.3)
		tmp = Float64(1.0 / exp(w));
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - w), $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 w < -1.30000000000000004

   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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1.30000000000000004 < w

   1. Initial program 99.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 13: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.18:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\log \ell, \ell, \ell\right), w, \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.18)
   (/ 1.0 (exp w))
   (if (<= w 0.115) (fma (fma (log l) l l) w l) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.18) {
		tmp = 1.0 / exp(w);
	} else if (w <= 0.115) {
		tmp = fma(fma(log(l), l, l), w, l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.18)
		tmp = Float64(1.0 / exp(w));
	elseif (w <= 0.115)
		tmp = fma(fma(log(l), l, l), w, l);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.18], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.115], N[(N[(N[Log[l], $MachinePrecision] * l + l), $MachinePrecision] * w + l), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.17999999999999999

   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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -0.17999999999999999 < w < 0.115000000000000005

   1. Initial program 99.2%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 95.5%

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

   if 0.115000000000000005 < 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: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;{\ell}^{1} \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68)
   (/ 1.0 (exp w))
   (if (<= w 0.115) (* (pow l 1.0) (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = 1.0 / exp(w);
	} else if (w <= 0.115) {
		tmp = pow(l, 1.0) * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.68d0)) then
        tmp = 1.0d0 / exp(w)
    else if (w <= 0.115d0) then
        tmp = (l ** 1.0d0) * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = 1.0 / Math.exp(w);
	} else if (w <= 0.115) {
		tmp = Math.pow(l, 1.0) * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = 1.0 / math.exp(w)
	elif w <= 0.115:
		tmp = math.pow(l, 1.0) * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = Float64(1.0 / exp(w));
	elseif (w <= 0.115)
		tmp = Float64((l ^ 1.0) * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = 1.0 / exp(w);
	elseif (w <= 0.115)
		tmp = (l ^ 1.0) * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[(1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.115], N[(N[Power[l, 1.0], $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.680000000000000049

   1. Initial program 100.0%

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

      1. 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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -0.680000000000000049 < w < 0.115000000000000005

   1. Initial program 99.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 95.0%

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

   if 0.115000000000000005 < 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. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;\frac{1}{e^{w}}\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;{\ell}^{1} \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;{\ell}^{1} \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68) (exp (- w)) (if (<= w 0.115) (* (pow l 1.0) (- 1.0 w)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = exp(-w);
	} else if (w <= 0.115) {
		tmp = pow(l, 1.0) * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.68d0)) then
        tmp = exp(-w)
    else if (w <= 0.115d0) then
        tmp = (l ** 1.0d0) * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = Math.exp(-w);
	} else if (w <= 0.115) {
		tmp = Math.pow(l, 1.0) * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = math.exp(-w)
	elif w <= 0.115:
		tmp = math.pow(l, 1.0) * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = exp(Float64(-w));
	elseif (w <= 0.115)
		tmp = Float64((l ^ 1.0) * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = exp(-w);
	elseif (w <= 0.115)
		tmp = (l ^ 1.0) * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 0.115], N[(N[Power[l, 1.0], $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.680000000000000049

   1. Initial program 100.0%

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

      1. 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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -0.680000000000000049 < w < 0.115000000000000005

   1. Initial program 99.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 95.0%

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

   if 0.115000000000000005 < 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. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.115:\\ \;\;\;\;{\ell}^{1} \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = 1.0 / exp(w);
	} else {
		tmp = pow(l, (1.0 + w)) * (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 = 1.0d0 / exp(w)
    else
        tmp = (l ** (1.0d0 + w)) * (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 = 1.0 / Math.exp(w);
	} else {
		tmp = Math.pow(l, (1.0 + w)) * (1.0 - w);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. 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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if -1 < w

   1. Initial program 99.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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 97.8

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

   8. Applied rewrites 97.8%

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

4. Final simplification 98.1%

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

Alternative 17: 46.0% 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 99.5%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 41.3

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

4. Applied rewrites 41.3%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-exp.f64 41.3

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

6. Applied rewrites 41.3%

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

Alternative 18: 16.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

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

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 18.8%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))