exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 18.4s

Alternatives: 11

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = 963.1534226377238
  2. l = -9.923477128223219e-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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

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

Alternative 2: 37.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   4. Applied rewrites 58.6%

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

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

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 37.3%

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

4. Final simplification 43.9%

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

Alternative 3: 33.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   4. Applied rewrites 58.6%

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

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

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 50.8

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

   4. Applied rewrites 50.8%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 29.9

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

   7. Applied rewrites 29.9%

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

4. Final simplification 38.7%

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

Alternative 4: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(w, l):
	tmp = 0
	if (math.pow(l, math.exp(w)) * 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((l ^ exp(w)) * exp(Float64(-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 (((l ^ exp(w)) * 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[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[Exp[(-w)], $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 99.6%

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

   4. Applied rewrites 58.6%

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

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

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 5.7

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

   7. Applied rewrites 5.7%

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

4. Final simplification 22.0%

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

Alternative 5: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(w, l):
	tmp = 0
	if (math.pow(l, math.exp(w)) * 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((l ^ exp(w)) * exp(Float64(-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 (((l ^ exp(w)) * 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[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[Exp[(-w)], $MachinePrecision]), $MachinePrecision], 1.1e-154], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   4. Applied rewrites 58.6%

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

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

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 4.7%

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

4. Final simplification 21.3%

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

Alternative 6: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.5:\\ \;\;\;\;e^{\frac{\left(-w\right) \cdot w}{w}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(e^{w}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.5) (exp (/ (* (- w) w) w)) (* 1.0 (pow l (exp w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4.5) {
		tmp = exp(((-w * w) / w));
	} else {
		tmp = 1.0 * pow(l, exp(w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-4.5d0)) then
        tmp = exp(((-w * w) / w))
    else
        tmp = 1.0d0 * (l ** exp(w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -4.5) {
		tmp = Math.exp(((-w * w) / w));
	} else {
		tmp = 1.0 * Math.pow(l, Math.exp(w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -4.5:
		tmp = math.exp(((-w * w) / w))
	else:
		tmp = 1.0 * math.pow(l, math.exp(w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4.5)
		tmp = exp(Float64(Float64(Float64(-w) * w) / w));
	else
		tmp = Float64(1.0 * (l ^ exp(w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -4.5)
		tmp = exp(((-w * w) / w));
	else
		tmp = 1.0 * (l ^ exp(w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4.5], N[Exp[N[(N[((-w) * w), $MachinePrecision] / w), $MachinePrecision]], $MachinePrecision], N[(1.0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4.5

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      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

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

      6. lift-exp.f64 99.2

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

   6. Applied rewrites 99.2%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -4.5 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.5:\\ \;\;\;\;e^{\frac{\left(-w\right) \cdot w}{w}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(e^{w}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.25:\\ \;\;\;\;e^{\frac{\left(-w\right) \cdot w}{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(w, 0.5, 1\right), w, 1\right)\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.25)
   (exp (/ (* (- w) w) w))
   (* (pow l (fma (fma w 0.5 1.0) w 1.0)) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.25) {
		tmp = exp(((-w * w) / w));
	} else {
		tmp = pow(l, fma(fma(w, 0.5, 1.0), w, 1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.25)
		tmp = exp(Float64(Float64(Float64(-w) * w) / w));
	else
		tmp = Float64((l ^ fma(fma(w, 0.5, 1.0), w, 1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.25], N[Exp[N[(N[((-w) * w), $MachinePrecision] / w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[(N[(w * 0.5 + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.25

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      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

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

      6. lift-exp.f64 99.2

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

   6. Applied rewrites 99.2%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -1.25 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around 0

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 80.8

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

   8. Applied rewrites 80.8%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.25:\\ \;\;\;\;e^{\frac{\left(-w\right) \cdot w}{w}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(w, 0.5, 1\right), w, 1\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.25:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(w, 0.5, 1\right), w, 1\right)\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.25) (exp (- w)) (* (pow l (fma (fma w 0.5 1.0) w 1.0)) 1.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.25) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, fma(fma(w, 0.5, 1.0), w, 1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.25)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ fma(fma(w, 0.5, 1.0), w, 1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.25], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(N[(w * 0.5 + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.25

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      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

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

      6. lift-exp.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -1.25 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around 0

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 80.8

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

   8. Applied rewrites 80.8%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.25:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(w, 0.5, 1\right), w, 1\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (1.0 + 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 ** (1.0d0 + 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, (1.0 + 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, (1.0 + 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 ^ Float64(1.0 + 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 ^ (1.0 + 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[(1.0 + w), $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

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      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

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

      6. lift-exp.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -1 < w

   1. Initial program 99.0%

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

   3. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.1

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 98.4%

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

4. Final simplification 98.7%

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

Alternative 10: 46.3% 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 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 52.8

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

4. Applied rewrites 52.8%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   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

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

   6. lift-exp.f64 52.8

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

6. Applied rewrites 52.8%

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

Alternative 11: 17.0% 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 99.3%

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

4. Applied rewrites 19.6%

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

Reproduce

herbie shell --seed 2024241 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))