exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 19.2s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. w = 6.704184316463337e-216
  2. l = -8.463140307208272e-71

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 36.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{-163}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 5e-163)
   0.0
   (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-163) {
		tmp = 0.0;
	} else {
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 5e-163)
		tmp = 0.0;
	else
		tmp = fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-163], 0.0, N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.99999999999999977e-163

   1. Initial program 100.0%

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

   4. Applied rewrites 65.6%

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

   if 4.99999999999999977e-163 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 35.6

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

   7. Applied rewrites 35.6%

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

Alternative 3: 32.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{-163}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w, w \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 5e-163) 0.0 (fma w (* w 0.5) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-163) {
		tmp = 0.0;
	} else {
		tmp = fma(w, (w * 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 5e-163)
		tmp = 0.0;
	else
		tmp = fma(w, Float64(w * 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-163], 0.0, N[(w * N[(w * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.99999999999999977e-163

   1. Initial program 100.0%

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

   4. Applied rewrites 65.6%

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

   if 4.99999999999999977e-163 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. 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 26.9

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 26.9%

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

Alternative 4: 31.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 2 \cdot 10^{-255}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 2e-255) 0.0 (* w (* w 0.5))))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 2e-255) {
		tmp = 0.0;
	} else {
		tmp = w * (w * 0.5);
	}
	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-255) then
        tmp = 0.0d0
    else
        tmp = w * (w * 0.5d0)
    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-255) {
		tmp = 0.0;
	} else {
		tmp = w * (w * 0.5);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (math.exp(-w) * math.pow(l, math.exp(w))) <= 2e-255:
		tmp = 0.0
	else:
		tmp = w * (w * 0.5)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 2e-255)
		tmp = 0.0;
	else
		tmp = Float64(w * Float64(w * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp(-w) * (l ^ exp(w))) <= 2e-255)
		tmp = 0.0;
	else
		tmp = w * (w * 0.5);
	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-255], 0.0, N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 2e-255

   1. Initial program 100.0%

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

   4. Applied rewrites 83.7%

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

   if 2e-255 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 43.8

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

   4. Applied rewrites 43.8%

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

   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 24.9

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 23.9%

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

Alternative 5: 19.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{-163}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (exp (- w)) (pow l (exp w))) 5e-163) 0.0 (- 1.0 w)))

C

double code(double w, double l) {
	double tmp;
	if ((exp(-w) * pow(l, exp(w))) <= 5e-163) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - w;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((exp(-w) * (l ** exp(w))) <= 5d-163) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - w
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((Math.exp(-w) * Math.pow(l, Math.exp(w))) <= 5e-163) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (math.exp(-w) * math.pow(l, math.exp(w))) <= 5e-163:
		tmp = 0.0
	else:
		tmp = 1.0 - w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (Float64(exp(Float64(-w)) * (l ^ exp(w))) <= 5e-163)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((exp(-w) * (l ^ exp(w))) <= 5e-163)
		tmp = 0.0;
	else
		tmp = 1.0 - w;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-163], 0.0, N[(1.0 - w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.99999999999999977e-163

   1. Initial program 100.0%

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

   4. Applied rewrites 65.6%

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

   if 4.99999999999999977e-163 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 5.8

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

   7. Applied rewrites 5.8%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -8.2 \cdot 10^{-16}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -8.2e-16)
   (exp (fma (log l) (exp w) (- w)))
   (* (pow l (exp w)) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -8.2e-16) {
		tmp = exp(fma(log(l), exp(w), -w));
	} else {
		tmp = pow(l, exp(w)) * 1.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -8.2e-16)
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	else
		tmp = Float64((l ^ exp(w)) * 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -8.2e-16], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -8.20000000000000012e-16

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in w around inf

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

      1. exp-to-pow N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

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

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

      13. log-rec N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. lower-exp.f64 N/A

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

      18. lower-neg.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if -8.20000000000000012e-16 < w

   1. Initial program 99.9%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -8.2 \cdot 10^{-16}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 18.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 65.6%

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

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

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 4.6%

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

Alternative 8: 98.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < 1

   1. Initial program 99.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 1 < l

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 99.6%

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

4. Final simplification 99.7%

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

Alternative 9: 97.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 125:\\ \;\;\;\;1 \cdot {\ell}^{1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68) (exp (- w)) (if (<= w 125.0) (* 1.0 (pow l 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = exp(-w);
	} else if (w <= 125.0) {
		tmp = 1.0 * pow(l, 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.68d0)) then
        tmp = exp(-w)
    else if (w <= 125.0d0) then
        tmp = 1.0d0 * (l ** 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = Math.exp(-w);
	} else if (w <= 125.0) {
		tmp = 1.0 * Math.pow(l, 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = math.exp(-w)
	elif w <= 125.0:
		tmp = 1.0 * math.pow(l, 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = exp(Float64(-w));
	elseif (w <= 125.0)
		tmp = Float64(1.0 * (l ^ 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = exp(-w);
	elseif (w <= 125.0)
		tmp = 1.0 * (l ^ 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 125.0], N[(1.0 * N[Power[l, 1.0], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.680000000000000049

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -0.680000000000000049 < w < 125

   1. Initial program 99.7%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 99.0%

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

   12. Taylor expanded in w around 0

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

   14. Applied rewrites 97.9%

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

   if 125 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

Alternative 10: 98.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -1 < w

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 99.3%

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

4. Final simplification 99.5%

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

Alternative 11: 40.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \mathsf{fma}\left(w, 0.5, -1\right)\\ \mathbf{if}\;w \leq -2 \cdot 10^{+154}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq -1.65 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \mathsf{fma}\left(w \cdot \left(w \cdot w\right), 0.125, -1\right), \frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.25, 0.5\right), 1\right)}, 1\right)\\ \mathbf{elif}\;w \leq 125:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0 \cdot t\_0, 1\right)}{1 + t\_0 \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (fma w 0.5 -1.0))))
   (if (<= w -2e+154)
     (* w (* w 0.5))
     (if (<= w -1.65e+77)
       (fma
        (* w (fma (* w (* w w)) 0.125 -1.0))
        (/ 1.0 (fma w (fma w 0.25 0.5) 1.0))
        1.0)
       (if (<= w 125.0)
         (/
          (fma t_0 (* t_0 t_0) 1.0)
          (+ 1.0 (* t_0 (fma w (fma w 0.5 -1.0) -1.0))))
         0.0)))))

C

double code(double w, double l) {
	double t_0 = w * fma(w, 0.5, -1.0);
	double tmp;
	if (w <= -2e+154) {
		tmp = w * (w * 0.5);
	} else if (w <= -1.65e+77) {
		tmp = fma((w * fma((w * (w * w)), 0.125, -1.0)), (1.0 / fma(w, fma(w, 0.25, 0.5), 1.0)), 1.0);
	} else if (w <= 125.0) {
		tmp = fma(t_0, (t_0 * t_0), 1.0) / (1.0 + (t_0 * fma(w, fma(w, 0.5, -1.0), -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = Float64(w * fma(w, 0.5, -1.0))
	tmp = 0.0
	if (w <= -2e+154)
		tmp = Float64(w * Float64(w * 0.5));
	elseif (w <= -1.65e+77)
		tmp = fma(Float64(w * fma(Float64(w * Float64(w * w)), 0.125, -1.0)), Float64(1.0 / fma(w, fma(w, 0.25, 0.5), 1.0)), 1.0);
	elseif (w <= 125.0)
		tmp = Float64(fma(t_0, Float64(t_0 * t_0), 1.0) / Float64(1.0 + Float64(t_0 * fma(w, fma(w, 0.5, -1.0), -1.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(w * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2e+154], N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -1.65e+77], N[(N[(w * N[(N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision] * 0.125 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(w * N[(w * 0.25 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[w, 125.0], N[(N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if w < -2.00000000000000007e154

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. 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 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto w \cdot \color{blue}{\left(w \cdot 0.5\right)} \]

   if -2.00000000000000007e154 < w < -1.6499999999999999e77

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. 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 6.6

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

   7. Applied rewrites 6.6%

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

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(w \cdot \left(w \cdot w\right), 0.125, -1\right) \cdot w, \color{blue}{\frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.25, 0.5\right), 1\right)}}, 1\right) \]

   if -1.6499999999999999e77 < w < 125

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 15.3

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

   4. Applied rewrites 15.3%

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

   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 5.2

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

   7. Applied rewrites 5.2%

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

   9. Applied rewrites 8.6%

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

   if 125 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+154}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq -1.65 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \mathsf{fma}\left(w \cdot \left(w \cdot w\right), 0.125, -1\right), \frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.25, 0.5\right), 1\right)}, 1\right)\\ \mathbf{elif}\;w \leq 125:\\ \;\;\;\;\frac{\mathsf{fma}\left(w \cdot \mathsf{fma}\left(w, 0.5, -1\right), \left(w \cdot \mathsf{fma}\left(w, 0.5, -1\right)\right) \cdot \left(w \cdot \mathsf{fma}\left(w, 0.5, -1\right)\right), 1\right)}{1 + \left(w \cdot \mathsf{fma}\left(w, 0.5, -1\right)\right) \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 52.3

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

4. Applied rewrites 52.3%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-exp.f64 52.3

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

6. Applied rewrites 52.3%

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

Alternative 13: 38.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+154}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 125:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \mathsf{fma}\left(w \cdot \left(w \cdot w\right), 0.125, -1\right), \frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.25, 0.5\right), 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2e+154)
   (* w (* w 0.5))
   (if (<= w 125.0)
     (fma
      (* w (fma (* w (* w w)) 0.125 -1.0))
      (/ 1.0 (fma w (fma w 0.25 0.5) 1.0))
      1.0)
     0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2e+154) {
		tmp = w * (w * 0.5);
	} else if (w <= 125.0) {
		tmp = fma((w * fma((w * (w * w)), 0.125, -1.0)), (1.0 / fma(w, fma(w, 0.25, 0.5), 1.0)), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2e+154)
		tmp = Float64(w * Float64(w * 0.5));
	elseif (w <= 125.0)
		tmp = fma(Float64(w * fma(Float64(w * Float64(w * w)), 0.125, -1.0)), Float64(1.0 / fma(w, fma(w, 0.25, 0.5), 1.0)), 1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2e+154], N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 125.0], N[(N[(w * N[(N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision] * 0.125 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(w * N[(w * 0.25 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -2.00000000000000007e154

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. 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 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto w \cdot \color{blue}{\left(w \cdot 0.5\right)} \]

   if -2.00000000000000007e154 < w < 125

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 27.8

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

   4. Applied rewrites 27.8%

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

   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 5.4

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

   7. Applied rewrites 5.4%

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

   9. Applied rewrites 19.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(w \cdot \left(w \cdot w\right), 0.125, -1\right) \cdot w, \color{blue}{\frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.25, 0.5\right), 1\right)}}, 1\right) \]

   if 125 < w

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+154}:\\ \;\;\;\;w \cdot \left(w \cdot 0.5\right)\\ \mathbf{elif}\;w \leq 125:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \mathsf{fma}\left(w \cdot \left(w \cdot w\right), 0.125, -1\right), \frac{1}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.25, 0.5\right), 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.3% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4 \cdot 10^{-310}:\\ \;\;\;\;w \cdot \mathsf{fma}\left(w, 0.5, -1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4e-310) (* w (fma w 0.5 -1.0)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4e-310) {
		tmp = w * fma(w, 0.5, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4e-310)
		tmp = Float64(w * fma(w, 0.5, -1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4e-310], N[(w * N[(w * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < -3.999999999999988e-310

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 55.8

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

   4. Applied rewrites 55.8%

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

   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 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 30.8%

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

   if -3.999999999999988e-310 < w

   1. Initial program 99.9%

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

   4. Applied rewrites 46.2%

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

Alternative 15: 16.7% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

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

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 20.3%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))