exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 20.6s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = -3761112.3563883794
  2. l = -1.2031982718521459e+157

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

Alternative 2: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ t_1 := t\_0 \cdot {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))) (t_1 (* t_0 (pow l (exp w)))))
   (if (<= t_1 0.0)
     0.0
     (if (<= t_1 1e+304)
       (* l (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0))
       t_0))))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double t_1 = t_0 * pow(l, exp(w));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 0.0;
	} else if (t_1 <= 1e+304) {
		tmp = l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	t_1 = Float64(t_0 * (l ^ exp(w)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = 0.0;
	elseif (t_1 <= 1e+304)
		tmp = Float64(l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], 0.0, If[LessEqual[t$95$1, 1e+304], N[(l * N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-up N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. mul0-lft N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. mul0-lft N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. associate-/r/ N/A

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

      21. div-inv N/A

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

      22. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e303

   1. Initial program 99.1%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.9

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

   8. Simplified 96.9%

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

   10. Applied egg-rr 97.4%

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

   11. Taylor expanded in w around 0

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

   13. Simplified 93.9%

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

   if 9.9999999999999994e303 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.6

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

   4. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 98.6

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

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{e^{-w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{+304}:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{+304}:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(w, 0.16666666666666666, 0.5\right), w \cdot w, w\right)\right)} \cdot \left(\ell - w \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* t_0 (pow l (exp w))) 1e+304)
     (* (pow l (fma (fma w 0.16666666666666666 0.5) (* w w) w)) (- l (* w l)))
     t_0)))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((t_0 * pow(l, exp(w))) <= 1e+304) {
		tmp = pow(l, fma(fma(w, 0.16666666666666666, 0.5), (w * w), w)) * (l - (w * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * (l ^ exp(w))) <= 1e+304)
		tmp = Float64((l ^ fma(fma(w, 0.16666666666666666, 0.5), Float64(w * w), w)) * Float64(l - Float64(w * l)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+304], N[(N[Power[l, N[(N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(w * w), $MachinePrecision] + w), $MachinePrecision]], $MachinePrecision] * N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e303

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 87.8

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

   5. Simplified 87.8%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 87.3

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

   8. Simplified 87.3%

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

   10. Applied egg-rr 87.8%

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

   11. Taylor expanded in w around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 97.5

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

   13. Simplified 97.5%

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

   if 9.9999999999999994e303 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.6

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

   4. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 98.6

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

   6. Applied egg-rr 98.6%

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

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{+304}:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(0.5, w \cdot w, w\right)\right)} \cdot \left(\ell \cdot \left(1 - w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* t_0 (pow l (exp w))) 1e+304)
     (* (pow l (fma 0.5 (* w w) w)) (* l (- 1.0 w)))
     t_0)))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((t_0 * pow(l, exp(w))) <= 1e+304) {
		tmp = pow(l, fma(0.5, (w * w), w)) * (l * (1.0 - w));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * (l ^ exp(w))) <= 1e+304)
		tmp = Float64((l ^ fma(0.5, Float64(w * w), w)) * Float64(l * Float64(1.0 - w)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+304], N[(N[Power[l, N[(0.5 * N[(w * w), $MachinePrecision] + w), $MachinePrecision]], $MachinePrecision] * N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e303

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 97.2

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

   5. Simplified 97.2%

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

   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 97.1

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

   8. Simplified 97.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. pow-plus N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 97.4

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

   10. Applied egg-rr 97.4%

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

   if 9.9999999999999994e303 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.6

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

   4. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 98.6

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

   6. Applied egg-rr 98.6%

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

Alternative 5: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{+304}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* t_0 (pow l (exp w))) 1e+304)
     (* (- 1.0 w) (pow l (fma w (fma w 0.5 1.0) 1.0)))
     t_0)))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((t_0 * pow(l, exp(w))) <= 1e+304) {
		tmp = (1.0 - w) * pow(l, fma(w, fma(w, 0.5, 1.0), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * (l ^ exp(w))) <= 1e+304)
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(w, fma(w, 0.5, 1.0), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+304], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e303

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 97.2

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

   5. Simplified 97.2%

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

   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 97.1

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

   8. Simplified 97.1%

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

   if 9.9999999999999994e303 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.6

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

   4. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 98.6

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

   6. Applied egg-rr 98.6%

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

Alternative 6: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot {\ell}^{\left(e^{w}\right)} \leq 10^{+304}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(w + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* t_0 (pow l (exp w))) 1e+304)
     (* (- 1.0 w) (pow l (+ w 1.0)))
     t_0)))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((t_0 * pow(l, exp(w))) <= 1e+304) {
		tmp = (1.0 - w) * pow(l, (w + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-w)
    if ((t_0 * (l ** exp(w))) <= 1d+304) then
        tmp = (1.0d0 - w) * (l ** (w + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.exp(-w);
	double tmp;
	if ((t_0 * Math.pow(l, Math.exp(w))) <= 1e+304) {
		tmp = (1.0 - w) * Math.pow(l, (w + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.exp(-w)
	tmp = 0
	if (t_0 * math.pow(l, math.exp(w))) <= 1e+304:
		tmp = (1.0 - w) * math.pow(l, (w + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * (l ^ exp(w))) <= 1e+304)
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(w + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = exp(-w);
	tmp = 0.0;
	if ((t_0 * (l ^ exp(w))) <= 1e+304)
		tmp = (1.0 - w) * (l ^ (w + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+304], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e303

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 97.2

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

   5. Simplified 97.2%

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

   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 96.6

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

   8. Simplified 96.6%

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

   if 9.9999999999999994e303 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.6

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

   4. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 98.6

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

   6. Applied egg-rr 98.6%

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

Alternative 7: 37.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 2e-156)
   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))) <= 2e-156) {
		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))) <= 2e-156)
		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], 2e-156], 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))) < 2.00000000000000008e-156

   1. Initial program 99.3%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-up N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. mul0-lft N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. mul0-lft N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. associate-/r/ N/A

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

      21. div-inv N/A

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

      22. metadata-eval N/A

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

   4. Applied egg-rr 45.8%

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

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

   1. Initial program 99.6%

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

      1. lift-exp.f64 N/A

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

      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 egg-rr 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 31.5

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

   7. Simplified 31.5%

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

Alternative 8: 32.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 2e-156)
   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))) <= 2e-156) {
		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))) <= 2e-156)
		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], 2e-156], 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))) < 2.00000000000000008e-156

   1. Initial program 99.3%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-up N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. mul0-lft N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. mul0-lft N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. associate-/r/ N/A

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

      21. div-inv N/A

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

      22. metadata-eval N/A

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

   4. Applied egg-rr 45.8%

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

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

   1. Initial program 99.6%

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

      1. lift-exp.f64 N/A

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

      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 egg-rr 43.7%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 25.9

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

   7. Simplified 25.9%

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

Alternative 9: 19.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp(-w) * (l ^ exp(w))) <= 2e-156)
		tmp = 0.0;
	else
		tmp = 1.0 - w;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-up N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. mul0-lft N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. mul0-lft N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. associate-/r/ N/A

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

      21. div-inv N/A

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

      22. metadata-eval N/A

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

   4. Applied egg-rr 45.8%

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

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

   1. Initial program 99.6%

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

      1. lift-exp.f64 N/A

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

      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 egg-rr 43.7%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 6.1

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

   7. Simplified 6.1%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.00055:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.00055)
   (exp (- (* (exp w) (log l)) w))
   (/
    (pow l (exp w))
    (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 <= -0.00055) {
		tmp = exp(((exp(w) * log(l)) - w));
	} else {
		tmp = pow(l, exp(w)) / 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 <= -0.00055)
		tmp = exp(Float64(Float64(exp(w) * log(l)) - w));
	else
		tmp = Float64((l ^ exp(w)) / 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, -0.00055], N[Exp[N[(N[(N[Exp[w], $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -5.50000000000000033e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.9

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

   5. Simplified 99.9%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

      1. lift-exp.f64 N/A

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

      2. pow-to-exp N/A

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

      3. div-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   9. Applied egg-rr 99.9%

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

   if -5.50000000000000033e-4 < w

   1. Initial program 99.3%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.1

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

   5. Simplified 99.1%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 98.3

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

   10. Simplified 98.3%

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

Alternative 11: 18.7% accurate, 1.0× speedup

Math

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

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 99.3%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-up N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. mul0-lft N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. mul0-lft N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. associate-/r/ N/A

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

      21. div-inv N/A

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

      22. metadata-eval N/A

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

   4. Applied egg-rr 45.8%

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

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

   1. Initial program 99.6%

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

      1. lift-exp.f64 N/A

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

      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 egg-rr 43.7%

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

   5. Taylor expanded in w around 0

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

   7. Simplified 5.0%

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

Alternative 12: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.58:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.58)
   (exp (- w))
   (/
    (pow l (exp w))
    (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 <= -1.58) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, exp(w)) / 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 <= -1.58)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / 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, -1.58], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.5800000000000001

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      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 egg-rr 100.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 100.0

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

   6. Applied egg-rr 100.0%

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

   if -1.5800000000000001 < w

   1. Initial program 99.3%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.1

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

   5. Simplified 99.1%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 98.0

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

   10. Simplified 98.0%

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

Alternative 13: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / 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 = (l ^ exp(w)) / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(w + 1.0), $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-exp.f64 N/A

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

      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 egg-rr 100.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. *-rgt-identity 100.0

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

   6. Applied egg-rr 100.0%

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

   if -1 < w

   1. Initial program 99.3%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.1

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

   5. Simplified 99.1%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.2

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

   10. Simplified 97.2%

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

Alternative 14: 91.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.98:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.98)
   (* l (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.98) {
		tmp = l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.98)
		tmp = Float64(l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.98], N[(l * N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.97999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 88.7

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

   5. Simplified 88.7%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 79.2

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

   8. Simplified 79.2%

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

   10. Applied egg-rr 79.5%

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

   11. Taylor expanded in w around 0

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

   13. Simplified 88.5%

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

   if 0.97999999999999998 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-up N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. mul0-lft N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. mul0-lft N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. associate-/r/ N/A

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

      21. div-inv N/A

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

      22. metadata-eval N/A

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

   4. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.98:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 16.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

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

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.5%

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

   1. exp-neg N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. mul0-lft N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. mul0-lft N/A

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

   14. metadata-eval N/A

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

   15. +-inverses N/A

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

   16. metadata-eval N/A

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

   17. flip-- N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. associate-/r/ N/A

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

   21. div-inv N/A

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

   22. metadata-eval N/A

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

4. Applied egg-rr 16.9%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))