exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 19.4s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. w = -1.3645817092798459e-39
  2. l = -2.8108054369938773e+257

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 70.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w} \cdot {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-308}:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* (exp (- w)) (pow l (exp w)))))
   (if (<= t_0 4e-308)
     0.0
     (if (<= t_0 INFINITY)
       (/ 1.0 (/ 1.0 l))
       (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0)))))

C

double code(double w, double l) {
	double t_0 = exp(-w) * pow(l, exp(w));
	double tmp;
	if (t_0 <= 4e-308) {
		tmp = 0.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 / (1.0 / l);
	} else {
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = Float64(exp(Float64(-w)) * (l ^ exp(w)))
	tmp = 0.0
	if (t_0 <= 4e-308)
		tmp = 0.0;
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 / Float64(1.0 / l));
	else
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-308], 0.0, If[LessEqual[t$95$0, Infinity], N[(1.0 / N[(1.0 / l), $MachinePrecision]), $MachinePrecision], N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.00000000000000013e-308

   1. Initial program 99.8%

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

   4. Applied rewrites 97.6%

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

   if 4.00000000000000013e-308 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < +inf.0

   1. Initial program 99.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\frac{-1}{\frac{-1}{\ell}}\right)}^{\left(e^{w}\right)}} \]
   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. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 68.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\ell}}} \]

   10. Applied rewrites 68.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\ell}}} \]

   if +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.7

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

   4. Applied rewrites 43.7%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 20.8

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

   7. Applied rewrites 20.8%

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

Alternative 3: 36.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{-155}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 5e-155)
   0.0
   (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-155) {
		tmp = 0.0;
	} else {
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 5e-155)
		tmp = 0.0;
	else
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 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], 5e-155], 0.0, N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   4. Applied rewrites 46.0%

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

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

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.5

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

   4. Applied rewrites 43.5%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 29.0

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

   7. Applied rewrites 29.0%

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

Alternative 4: 32.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{-155}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 5e-155)
   0.0
   (fma w (fma w 0.5 -1.0) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-155) {
		tmp = 0.0;
	} else {
		tmp = fma(w, fma(w, 0.5, -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 5e-155)
		tmp = 0.0;
	else
		tmp = fma(w, fma(w, 0.5, -1.0), 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], 5e-155], 0.0, N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   4. Applied rewrites 46.0%

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

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

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.5

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

   4. Applied rewrites 43.5%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

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

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

      9. distribute-rgt-in N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-lft-in N/A

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

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

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 25.1

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

   7. Applied rewrites 25.1%

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

Alternative 5: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-155) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - 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))) <= 5d-155) 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))) <= 5e-155) {
		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))) <= 5e-155:
		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))) <= 5e-155)
		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))) <= 5e-155)
		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], 5e-155], 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))) < 4.9999999999999999e-155

   1. Initial program 99.8%

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

   4. Applied rewrites 46.0%

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

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

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.5

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

   4. Applied rewrites 43.5%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 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 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -7.6 \cdot 10^{-12}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.16666666666666666, 0.5\right), 1\right), 1\right)}{{\ell}^{\left(e^{w}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -7.6e-12)
   (exp (fma (log l) (exp w) (- w)))
   (/
    1.0
    (/
     (fma w (fma w (fma w 0.16666666666666666 0.5) 1.0) 1.0)
     (pow l (exp w))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -7.6e-12) {
		tmp = exp(fma(log(l), exp(w), -w));
	} else {
		tmp = 1.0 / (fma(w, fma(w, fma(w, 0.16666666666666666, 0.5), 1.0), 1.0) / pow(l, exp(w)));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -7.6e-12)
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	else
		tmp = Float64(1.0 / Float64(fma(w, fma(w, fma(w, 0.16666666666666666, 0.5), 1.0), 1.0) / (l ^ exp(w))));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -7.6e-12], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -7.59999999999999993e-12

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\frac{-1}{\frac{-1}{\ell}}\right)}^{\left(e^{w}\right)}} \]
   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. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in l around inf

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

      1. div-exp N/A

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

      2. lower-exp.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. log-rec N/A

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

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

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if -7.59999999999999993e-12 < w

   1. Initial program 99.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\frac{-1}{\frac{-1}{\ell}}\right)}^{\left(e^{w}\right)}} \]
   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. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.2

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

   10. Applied rewrites 99.2%

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

Alternative 7: 18.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 1.12e-154) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((exp(-w) * (l ** exp(w))) <= 1.12d-154) then
        tmp = 0.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((Math.exp(-w) * Math.pow(l, Math.exp(w))) <= 1.12e-154) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (math.exp(-w) * math.pow(l, math.exp(w))) <= 1.12e-154:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 1.12e-154)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp(-w) * (l ^ exp(w))) <= 1.12e-154)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.12e-154], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   4. Applied rewrites 46.0%

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

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

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.5

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

   4. Applied rewrites 43.5%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 4.8%

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

Alternative 8: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[Exp[(-w)], $MachinePrecision], N[(1.0 / N[(N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Power[l, N[Exp[w], $MachinePrecision]], $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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\frac{-1}{\frac{-1}{\ell}}\right)}^{\left(e^{w}\right)}} \]
   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. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.1

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

   10. Applied rewrites 99.1%

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

Alternative 9: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.2:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)}{{\ell}^{\left(e^{w}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.2)
   (exp (- w))
   (/ 1.0 (/ (fma w (fma w 0.5 1.0) 1.0) (pow l (exp w))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4.2) {
		tmp = exp(-w);
	} else {
		tmp = 1.0 / (fma(w, fma(w, 0.5, 1.0), 1.0) / pow(l, exp(w)));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4.2)
		tmp = exp(Float64(-w));
	else
		tmp = Float64(1.0 / Float64(fma(w, fma(w, 0.5, 1.0), 1.0) / (l ^ exp(w))));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4.2], N[Exp[(-w)], $MachinePrecision], N[(1.0 / N[(N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4.20000000000000018

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -4.20000000000000018 < w

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\frac{-1}{\frac{-1}{\ell}}\right)}^{\left(e^{w}\right)}} \]
   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. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.0

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

   10. Applied rewrites 99.0%

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

Alternative 10: 98.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[Exp[(-w)], $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in w around 0

      \[\leadsto \left(1 - w\right) \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right)\right)}} \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 98.9

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

   8. Applied rewrites 98.9%

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

Alternative 11: 98.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[Exp[(-w)], $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if 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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1.30000000000000004 < w

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\left(1 + -1 \cdot w\right)} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in w around 0

      \[\leadsto \left(1 - w\right) \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}} \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.9

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

   8. Applied rewrites 98.9%

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

Alternative 12: 97.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.135:\\ \;\;\;\;1 \cdot {\ell}^{1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7) (exp (- w)) (if (<= w 0.135) (* 1.0 (pow l 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp(-w);
	} else if (w <= 0.135) {
		tmp = 1.0 * pow(l, 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.7d0)) then
        tmp = exp(-w)
    else if (w <= 0.135d0) then
        tmp = 1.0d0 * (l ** 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = Math.exp(-w);
	} else if (w <= 0.135) {
		tmp = 1.0 * Math.pow(l, 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = math.exp(-w)
	elif w <= 0.135:
		tmp = 1.0 * math.pow(l, 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(-w));
	elseif (w <= 0.135)
		tmp = Float64(1.0 * (l ^ 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.7)
		tmp = exp(-w);
	elseif (w <= 0.135)
		tmp = 1.0 * (l ^ 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 0.135], N[(1.0 * N[Power[l, 1.0], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -0.69999999999999996 < w < 0.13500000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\left(1 + -1 \cdot w\right)} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   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 97.4%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 97.4%

       \[\leadsto 1 \cdot {\ell}^{1} \]

   if 0.13500000000000001 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 13: 98.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -1 < w

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.7

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

   8. Applied rewrites 98.7%

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

Alternative 14: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.135:\\ \;\;\;\;\frac{1}{\frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7) (exp (- w)) (if (<= w 0.135) (/ 1.0 (/ 1.0 l)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp(-w);
	} else if (w <= 0.135) {
		tmp = 1.0 / (1.0 / l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.7d0)) then
        tmp = exp(-w)
    else if (w <= 0.135d0) then
        tmp = 1.0d0 / (1.0d0 / l)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = Math.exp(-w);
	} else if (w <= 0.135) {
		tmp = 1.0 / (1.0 / l);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = math.exp(-w)
	elif w <= 0.135:
		tmp = 1.0 / (1.0 / l)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(-w));
	elseif (w <= 0.135)
		tmp = Float64(1.0 / Float64(1.0 / l));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.7)
		tmp = exp(-w);
	elseif (w <= 0.135)
		tmp = 1.0 / (1.0 / l);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 0.135], N[(1.0 / N[(1.0 / l), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

   6. Applied rewrites 100.0%

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

   if -0.69999999999999996 < w < 0.13500000000000001

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

      \[\leadsto e^{-w} \cdot \color{blue}{{\left(\frac{-1}{\frac{-1}{\ell}}\right)}^{\left(e^{w}\right)}} \]
   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. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 97.2

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\ell}}} \]

   10. Applied rewrites 97.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\ell}}} \]

   if 0.13500000000000001 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 15: 16.3% 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 99.8%

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

4. Applied rewrites 16.1%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))