Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 97.7%

Time: 14.9s

Alternatives: 16

Speedup: 1.0×

• 21.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

Wolfram

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

Wolfram

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+270)
     (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+270) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 2d+270) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+270) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+270:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+270)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+270)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+270], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.0000000000000001e270

   1. Initial program 98.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto a \cdot \color{blue}{\frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)} \cdot \color{blue}{a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\right), \color{blue}{a}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({k}^{m}\right), \left(1 + k \cdot \left(k + 10\right)\right)\right), a\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \left(1 + k \cdot \left(k + 10\right)\right)\right), a\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \left(k \cdot \left(k + 10\right)\right)\right)\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right), a\right) \]

      8. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, 10\right)\right)\right)\right), a\right) \]

   6. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)} \cdot a} \]

   if 2.0000000000000001e270 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 60.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 60.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({k}^{m}\right)}\right) \]

      2. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, \color{blue}{m}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -2.5 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -2.5e-10)
     t_0
     (if (<= m 2.2e-15) (/ a (+ 1.0 (* k (+ k 10.0)))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -2.5e-10) {
		tmp = t_0;
	} else if (m <= 2.2e-15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-2.5d-10)) then
        tmp = t_0
    else if (m <= 2.2d-15) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -2.5e-10) {
		tmp = t_0;
	} else if (m <= 2.2e-15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -2.5e-10:
		tmp = t_0
	elif m <= 2.2e-15:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -2.5e-10)
		tmp = t_0;
	elseif (m <= 2.2e-15)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -2.5e-10)
		tmp = t_0;
	elseif (m <= 2.2e-15)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.5e-10], t$95$0, If[LessEqual[m, 2.2e-15], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.50000000000000016e-10 or 2.19999999999999986e-15 < m

   1. Initial program 87.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 87.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({k}^{m}\right)}\right) \]

      2. pow-lowering-pow.f64 98.5%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, \color{blue}{m}\right)\right) \]

   7. Simplified 98.5%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   if -2.50000000000000016e-10 < m < 2.19999999999999986e-15

   1. Initial program 94.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7.5e-14) (* a (pow k m)) (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7.5e-14) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m - 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 7.5d-14) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7.5e-14) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7.5e-14:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7.5e-14)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7.5e-14)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7.5e-14], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.4999999999999996e-14

   1. Initial program 92.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 92.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({k}^{m}\right)}\right) \]

      2. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, \color{blue}{m}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   if 7.4999999999999996e-14 < k

   1. Initial program 83.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 83.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}\right), \left({k}^{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot e^{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}\right), \left({k}^{2}\right)\right) \]

      4. exp-prod N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}\right), \left({k}^{2}\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {\left(e^{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}\right)}^{m}\right), \left({k}^{2}\right)\right) \]

      6. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {\left(e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log k\right)\right)\right)}\right)}^{m}\right), \left({k}^{2}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {\left(e^{\log k}\right)}^{m}\right), \left({k}^{2}\right)\right) \]

      8. rem-exp-log N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \left({k}^{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   7. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto a \cdot \color{blue}{\frac{{k}^{m}}{k \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{k}^{m}}{k \cdot k} \cdot \color{blue}{a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{k}^{m}}{k \cdot k}\right), \color{blue}{a}\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{k}^{m}}{{k}^{2}}\right), a\right) \]

      5. pow-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left({k}^{\left(m - 2\right)}\right), a\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \left(m - 2\right)\right), a\right) \]

      7. --lowering--.f64 95.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \mathsf{\_.f64}\left(m, 2\right)\right), a\right) \]

   9. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{{k}^{\left(m - 2\right)} \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.15:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{1 + \frac{1}{k \cdot k} \cdot \left(\frac{1000}{k} + -100\right)}{k} \cdot \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (/ (- a (* (/ a k) (+ 10.0 (/ -99.0 k)))) (* k k))
   (if (<= m 0.15)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/
      a
      (+
       1.0
       (*
        (/ (+ 1.0 (* (/ 1.0 (* k k)) (+ (/ 1000.0 k) -100.0))) k)
        (/ k (/ 1.0 (+ (* k k) (- 100.0 (* k -10.0)))))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (1.0 + (((1.0 + ((1.0 / (k * k)) * ((1000.0 / k) + -100.0))) / k) * (k / (1.0 / ((k * k) + (100.0 - (k * -10.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = (a - ((a / k) * (10.0d0 + ((-99.0d0) / k)))) / (k * k)
    else if (m <= 0.15d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a / (1.0d0 + (((1.0d0 + ((1.0d0 / (k * k)) * ((1000.0d0 / k) + (-100.0d0)))) / k) * (k / (1.0d0 / ((k * k) + (100.0d0 - (k * (-10.0d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (1.0 + (((1.0 + ((1.0 / (k * k)) * ((1000.0 / k) + -100.0))) / k) * (k / (1.0 / ((k * k) + (100.0 - (k * -10.0)))))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k)
	elif m <= 0.15:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a / (1.0 + (((1.0 + ((1.0 / (k * k)) * ((1000.0 / k) + -100.0))) / k) * (k / (1.0 / ((k * k) + (100.0 - (k * -10.0)))))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(Float64(a - Float64(Float64(a / k) * Float64(10.0 + Float64(-99.0 / k)))) / Float64(k * k));
	elseif (m <= 0.15)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a / Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(k * k)) * Float64(Float64(1000.0 / k) + -100.0))) / k) * Float64(k / Float64(1.0 / Float64(Float64(k * k) + Float64(100.0 - Float64(k * -10.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	elseif (m <= 0.15)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a / (1.0 + (((1.0 + ((1.0 / (k * k)) * ((1000.0 / k) + -100.0))) / k) * (k / (1.0 / ((k * k) + (100.0 - (k * -10.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(N[(a - N[(N[(a / k), $MachinePrecision] * N[(10.0 + N[(-99.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.15], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(N[(N[(1.0 + N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1000.0 / k), $MachinePrecision] + -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(k / N[(1.0 / N[(N[(k * k), $MachinePrecision] + N[(100.0 - N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 73.0%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 13.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 13.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   12. Simplified 65.7%

       \[\leadsto \color{blue}{\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} + 10\right)}{k \cdot k}} \]

   if -1.12e8 < m < 0.149999999999999994

   1. Initial program 94.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 0.149999999999999994 < m

   1. Initial program 74.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{k \cdot k - 10 \cdot 10}{k - 10} \cdot k\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\color{blue}{k - 10}}\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{k + -10}\right)\right)\right) \]

      6. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\frac{{k}^{3} + {-10}^{3}}{\color{blue}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}}\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left({k}^{3} + {-10}^{3}\right) \cdot \color{blue}{\frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}}\right)\right)\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) + {-10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) + -1000\right) \cdot \frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) + \left(\mathsf{neg}\left(1000\right)\right)\right) \cdot \frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) - 1000\right) \cdot \frac{\color{blue}{1}}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      12. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{k \cdot k - 10 \cdot 10}{k \cdot \left(k \cdot k\right) - 1000} \cdot \color{blue}{\frac{k}{\frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{k \cdot k - 10 \cdot 10}{k \cdot \left(k \cdot k\right) - 1000}\right), \color{blue}{\left(\frac{k}{\frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)}\right)\right)\right) \]

   9. Applied egg-rr 2.4%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k \cdot k + -100}{k \cdot \left(k \cdot k\right) + -1000} \cdot \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1000}{{k}^{3}}\right) - 100 \cdot \frac{1}{{k}^{2}}}{k}\right)}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1000}{{k}^{3}}\right) - 100 \cdot \frac{1}{{k}^{2}}\right), k\right), \mathsf{/.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 37.3%

       \[\leadsto \frac{a}{1 + \color{blue}{\frac{1 + \frac{1}{k \cdot k} \cdot \left(\frac{1000}{k} + -100\right)}{k}} \cdot \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.15:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{1 + \frac{1}{k \cdot k} \cdot \left(\frac{1000}{k} + -100\right)}{k} \cdot \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}} \cdot \frac{1 + \frac{-100}{k \cdot k}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (/ (- a (* (/ a k) (+ 10.0 (/ -99.0 k)))) (* k k))
   (if (<= m 0.28)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/
      a
      (+
       1.0
       (*
        (/ k (/ 1.0 (+ (* k k) (- 100.0 (* k -10.0)))))
        (/ (+ 1.0 (/ -100.0 (* k k))) k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.28) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (1.0 + ((k / (1.0 / ((k * k) + (100.0 - (k * -10.0))))) * ((1.0 + (-100.0 / (k * k))) / k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = (a - ((a / k) * (10.0d0 + ((-99.0d0) / k)))) / (k * k)
    else if (m <= 0.28d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a / (1.0d0 + ((k / (1.0d0 / ((k * k) + (100.0d0 - (k * (-10.0d0)))))) * ((1.0d0 + ((-100.0d0) / (k * k))) / k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.28) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (1.0 + ((k / (1.0 / ((k * k) + (100.0 - (k * -10.0))))) * ((1.0 + (-100.0 / (k * k))) / k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k)
	elif m <= 0.28:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a / (1.0 + ((k / (1.0 / ((k * k) + (100.0 - (k * -10.0))))) * ((1.0 + (-100.0 / (k * k))) / k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(Float64(a - Float64(Float64(a / k) * Float64(10.0 + Float64(-99.0 / k)))) / Float64(k * k));
	elseif (m <= 0.28)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a / Float64(1.0 + Float64(Float64(k / Float64(1.0 / Float64(Float64(k * k) + Float64(100.0 - Float64(k * -10.0))))) * Float64(Float64(1.0 + Float64(-100.0 / Float64(k * k))) / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	elseif (m <= 0.28)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a / (1.0 + ((k / (1.0 / ((k * k) + (100.0 - (k * -10.0))))) * ((1.0 + (-100.0 / (k * k))) / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(N[(a - N[(N[(a / k), $MachinePrecision] * N[(10.0 + N[(-99.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.28], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(N[(k / N[(1.0 / N[(N[(k * k), $MachinePrecision] + N[(100.0 - N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-100.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 73.0%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 13.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 13.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   12. Simplified 65.7%

       \[\leadsto \color{blue}{\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} + 10\right)}{k \cdot k}} \]

   if -1.12e8 < m < 0.28000000000000003

   1. Initial program 94.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 0.28000000000000003 < m

   1. Initial program 74.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{k \cdot k - 10 \cdot 10}{k - 10} \cdot k\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\color{blue}{k - 10}}\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{k + -10}\right)\right)\right) \]

      6. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\frac{{k}^{3} + {-10}^{3}}{\color{blue}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}}\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left({k}^{3} + {-10}^{3}\right) \cdot \color{blue}{\frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}}\right)\right)\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) + {-10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) + -1000\right) \cdot \frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) + \left(\mathsf{neg}\left(1000\right)\right)\right) \cdot \frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\left(k \cdot \left(k \cdot k\right) - 1000\right) \cdot \frac{\color{blue}{1}}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)\right)\right) \]

      12. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{k \cdot k - 10 \cdot 10}{k \cdot \left(k \cdot k\right) - 1000} \cdot \color{blue}{\frac{k}{\frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{k \cdot k - 10 \cdot 10}{k \cdot \left(k \cdot k\right) - 1000}\right), \color{blue}{\left(\frac{k}{\frac{1}{k \cdot k + \left(-10 \cdot -10 - k \cdot -10\right)}}\right)}\right)\right)\right) \]

   9. Applied egg-rr 2.4%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k \cdot k + -100}{k \cdot \left(k \cdot k\right) + -1000} \cdot \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left(\frac{1 - 100 \cdot \frac{1}{{k}^{2}}}{k}\right)}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - 100 \cdot \frac{1}{{k}^{2}}\right), k\right), \mathsf{/.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(100 \cdot \frac{1}{{k}^{2}}\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(100 \cdot \frac{1}{{k}^{2}}\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{100 \cdot 1}{{k}^{2}}\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{100}{{k}^{2}}\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(100\right)}{{k}^{2}}\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-100}{{k}^{2}}\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-100, \left({k}^{2}\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-100, \left(k \cdot k\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 35.7%

           \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-100, \mathsf{*.f64}\left(k, k\right)\right)\right), k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 35.7%

       \[\leadsto \frac{a}{1 + \color{blue}{\frac{1 + \frac{-100}{k \cdot k}}{k}} \cdot \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{k}{\frac{1}{k \cdot k + \left(100 - k \cdot -10\right)}} \cdot \frac{1 + \frac{-100}{k \cdot k}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k \cdot \left(1 + \frac{10 + \frac{-1000}{k \cdot k}}{k}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (/ (- a (* (/ a k) (+ 10.0 (/ -99.0 k)))) (* k k))
   (if (<= m 0.9)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/ a (+ 1.0 (* k (* k (+ 1.0 (/ (+ 10.0 (/ -1000.0 (* k k))) k)))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.9) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (1.0 + (k * (k * (1.0 + ((10.0 + (-1000.0 / (k * k))) / k)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = (a - ((a / k) * (10.0d0 + ((-99.0d0) / k)))) / (k * k)
    else if (m <= 0.9d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a / (1.0d0 + (k * (k * (1.0d0 + ((10.0d0 + ((-1000.0d0) / (k * k))) / k)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.9) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (1.0 + (k * (k * (1.0 + ((10.0 + (-1000.0 / (k * k))) / k)))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k)
	elif m <= 0.9:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a / (1.0 + (k * (k * (1.0 + ((10.0 + (-1000.0 / (k * k))) / k)))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(Float64(a - Float64(Float64(a / k) * Float64(10.0 + Float64(-99.0 / k)))) / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k * Float64(1.0 + Float64(Float64(10.0 + Float64(-1000.0 / Float64(k * k))) / k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	elseif (m <= 0.9)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a / (1.0 + (k * (k * (1.0 + ((10.0 + (-1000.0 / (k * k))) / k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(N[(a - N[(N[(a / k), $MachinePrecision] * N[(10.0 + N[(-99.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k * N[(1.0 + N[(N[(10.0 + N[(-1000.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 73.0%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 13.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 13.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   12. Simplified 65.7%

       \[\leadsto \color{blue}{\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} + 10\right)}{k \cdot k}} \]

   if -1.12e8 < m < 0.900000000000000022

   1. Initial program 94.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 0.900000000000000022 < m

   1. Initial program 74.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 68.9%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 2.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 2.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left({k}^{4}\right)}, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({k}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({k}^{3} \cdot k\right), \mathsf{+.f64}\left(\color{blue}{100}, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot {k}^{3}\right), \mathsf{+.f64}\left(\color{blue}{100}, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3}\right)\right), \mathsf{+.f64}\left(\color{blue}{100}, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot {k}^{2}\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2}\right)\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 2.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, -10\right)\right)\right)\right)\right)\right) \]

   12. Simplified 2.5%

       \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{100 + k \cdot \left(k + -10\right)}} \]

   13. Taylor expanded in k around -inf

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + -1 \cdot \frac{1000 \cdot \frac{1}{{k}^{2}} - 10}{k}\right)\right)}\right)\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + -1 \cdot \frac{1000 \cdot \frac{1}{{k}^{2}} - 10}{k}\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + -1 \cdot \frac{1000 \cdot \frac{1}{{k}^{2}} - 10}{k}\right)\right)}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + -1 \cdot \frac{1000 \cdot \frac{1}{{k}^{2}} - 10}{k}\right)\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(1 + -1 \cdot \frac{1000 \cdot \frac{1}{{k}^{2}} - 10}{k}\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1000 \cdot \frac{1}{{k}^{2}} - 10}{k}\right)}\right)\right)\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(1000 \cdot \frac{1}{{k}^{2}} - 10\right)}{\color{blue}{k}}\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(1000 \cdot \frac{1}{{k}^{2}} - 10\right)\right), \color{blue}{k}\right)\right)\right)\right)\right)\right) \]

   15. Simplified 35.0%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(k \cdot \left(1 + \frac{10 + \frac{-1000}{k \cdot k}}{k}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k \cdot \left(1 + \frac{10 + \frac{-1000}{k \cdot k}}{k}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.086:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(-10 - k \cdot -99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (/ (- a (* (/ a k) (+ 10.0 (/ -99.0 k)))) (* k k))
   (if (<= m 0.086)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* a (- -10.0 (* k -99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.086) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (-10.0 - (k * -99.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = (a - ((a / k) * (10.0d0 + ((-99.0d0) / k)))) / (k * k)
    else if (m <= 0.086d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * ((-10.0d0) - (k * (-99.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	} else if (m <= 0.086) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (-10.0 - (k * -99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k)
	elif m <= 0.086:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (-10.0 - (k * -99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(Float64(a - Float64(Float64(a / k) * Float64(10.0 + Float64(-99.0 / k)))) / Float64(k * k));
	elseif (m <= 0.086)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(-10.0 - Float64(k * -99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = (a - ((a / k) * (10.0 + (-99.0 / k)))) / (k * k);
	elseif (m <= 0.086)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (-10.0 - (k * -99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(N[(a - N[(N[(a / k), $MachinePrecision] * N[(10.0 + N[(-99.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.086], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(-10.0 - N[(k * -99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 73.0%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 13.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 13.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   12. Simplified 65.7%

       \[\leadsto \color{blue}{\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} + 10\right)}{k \cdot k}} \]

   if -1.12e8 < m < 0.085999999999999993

   1. Initial program 94.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 0.085999999999999993 < m

   1. Initial program 74.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 69.2%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 2.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 2.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(10\right)\right) \cdot a}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + -10 \cdot a\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \left(-1 \cdot k\right) \cdot \color{blue}{\left(a + -100 \cdot a\right)}\right)\right)\right) \]

       7. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \left(-1 \cdot k\right) \cdot \left(\left(-100 + 1\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \left(\left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right) \cdot \color{blue}{a}\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(-10 + \left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 + \left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right)}\right)\right)\right) \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \left(-10 + \left(\mathsf{neg}\left(k\right)\right) \cdot \left(\color{blue}{-100} + 1\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \left(-10 + \left(\mathsf{neg}\left(k \cdot \left(-100 + 1\right)\right)\right)\right)\right)\right)\right) \]

       13. unsub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \left(-10 - \color{blue}{k \cdot \left(-100 + 1\right)}\right)\right)\right)\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(-10, \color{blue}{\left(k \cdot \left(-100 + 1\right)\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(-10, \mathsf{*.f64}\left(k, \color{blue}{\left(-100 + 1\right)}\right)\right)\right)\right)\right) \]

       16. metadata-eval 25.3%

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(-10, \mathsf{*.f64}\left(k, -99\right)\right)\right)\right)\right) \]

   12. Simplified 25.3%

       \[\leadsto \color{blue}{a + k \cdot \left(a \cdot \left(-10 - k \cdot -99\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(10 + \frac{-99}{k}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.086:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(-10 - k \cdot -99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.086:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(-10 - k \cdot -99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (* a (/ 1.0 (* k k)))
   (if (<= m 0.086)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* a (- -10.0 (* k -99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 0.086) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (-10.0 - (k * -99.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 0.086d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * ((-10.0d0) - (k * (-99.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 0.086) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (-10.0 - (k * -99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = a * (1.0 / (k * k))
	elif m <= 0.086:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (-10.0 - (k * -99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 0.086)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(-10.0 - Float64(k * -99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 0.086)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (-10.0 - (k * -99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.086], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(-10.0 - N[(k * -99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -1.12e8 < m < 0.085999999999999993

   1. Initial program 94.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 0.085999999999999993 < m

   1. Initial program 74.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\frac{{k}^{3} + {10}^{3}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left({k}^{3} + {10}^{3}\right) \cdot \frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right) \cdot k\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(\left({k}^{3} + {10}^{3}\right) \cdot \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{3} + {10}^{3}\right), \color{blue}{\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{3}\right), \left({10}^{3}\right)\right), \left(\color{blue}{\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \left({10}^{3}\right)\right), \left(\frac{\color{blue}{1}}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \left({10}^{3}\right)\right), \left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)} \cdot k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \left(\frac{1}{\color{blue}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}} \cdot k\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right), \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k + \left(10 \cdot 10 - k \cdot 10\right)}\right), \color{blue}{k}\right)\right)\right)\right) \]

   6. Applied egg-rr 69.2%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{\left(k \cdot \left(k \cdot k\right) + 1000\right) \cdot \left(\frac{1}{100 + k \cdot \left(k + -10\right)} \cdot k\right)}} \]

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{k \cdot \left(1000 + {k}^{3}\right)}{100 + k \cdot \left(k - 10\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left(1000 + {k}^{3}\right)\right), \color{blue}{\left(100 + k \cdot \left(k - 10\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(1000 + {k}^{3}\right)\right), \left(\color{blue}{100} + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3} + 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left({k}^{3}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(k \cdot {k}^{2}\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2}\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \left(100 + k \cdot \left(k - 10\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(k \cdot \left(k - 10\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \color{blue}{\left(k - 10\right)}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \left(k + -10\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 2.5%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), 1000\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{-10}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 2.5%

      \[\leadsto \color{blue}{\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot k\right) + 1000\right)}{100 + k \cdot \left(k + -10\right)}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(10\right)\right) \cdot a}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + -10 \cdot a\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \left(-1 \cdot k\right) \cdot \color{blue}{\left(a + -100 \cdot a\right)}\right)\right)\right) \]

       7. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \left(-1 \cdot k\right) \cdot \left(\left(-100 + 1\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-10 \cdot a + \left(\left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right) \cdot \color{blue}{a}\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(-10 + \left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 + \left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right)}\right)\right)\right) \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \left(-10 + \left(\mathsf{neg}\left(k\right)\right) \cdot \left(\color{blue}{-100} + 1\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \left(-10 + \left(\mathsf{neg}\left(k \cdot \left(-100 + 1\right)\right)\right)\right)\right)\right)\right) \]

       13. unsub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \left(-10 - \color{blue}{k \cdot \left(-100 + 1\right)}\right)\right)\right)\right) \]

       14. --lowering--.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(-10, \color{blue}{\left(k \cdot \left(-100 + 1\right)\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(-10, \mathsf{*.f64}\left(k, \color{blue}{\left(-100 + 1\right)}\right)\right)\right)\right)\right) \]

       16. metadata-eval 25.3%

           \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(-10, \mathsf{*.f64}\left(k, -99\right)\right)\right)\right)\right) \]

   12. Simplified 25.3%

       \[\leadsto \color{blue}{a + k \cdot \left(a \cdot \left(-10 - k \cdot -99\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.086:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(-10 - k \cdot -99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (* a (/ 1.0 (* k k)))
   (if (<= m 4.4e+18) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* a (* k -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 4.4e+18) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 4.4d+18) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 4.4e+18) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = a * (1.0 / (k * k))
	elif m <= 4.4e+18:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 4.4e+18)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 4.4e+18)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.4e+18], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -1.12e8 < m < 4.4e18

   1. Initial program 95.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 95.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 4.4e18 < m

   1. Initial program 74.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(-10 \cdot \left(a \cdot k\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot k\right) \cdot \color{blue}{-10}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \left(-10 \cdot \color{blue}{k}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

      7. *-lowering-*.f64 7.5%

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   10. Simplified 7.5%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -112000000.0)
   (* a (/ 1.0 (* k k)))
   (if (<= m 5.9e+18) (/ a (+ 1.0 (* k k))) (+ a (* a (* k -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 5.9e+18) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (a * (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-112000000.0d0)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 5.9d+18) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a + (a * (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -112000000.0) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 5.9e+18) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (a * (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -112000000.0:
		tmp = a * (1.0 / (k * k))
	elif m <= 5.9e+18:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a + (a * (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -112000000.0)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 5.9e+18)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a + Float64(a * Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -112000000.0)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 5.9e+18)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a + (a * (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -112000000.0], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.9e+18], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.12e8

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -1.12e8 < m < 5.9e18

   1. Initial program 95.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 95.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      2. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   10. Simplified 85.6%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}} \]

   if 5.9e18 < m

   1. Initial program 74.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(-10 \cdot \left(a \cdot k\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot k\right) \cdot \color{blue}{-10}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \left(-10 \cdot \color{blue}{k}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

      7. *-lowering-*.f64 7.5%

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   10. Simplified 7.5%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -112000000:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 235000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -4.7e-303)
   (* a (/ 1.0 (* k k)))
   (if (<= k 235000.0) (/ a (+ 1.0 (* k 10.0))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.7e-303) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 235000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-4.7d-303)) then
        tmp = a * (1.0d0 / (k * k))
    else if (k <= 235000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.7e-303) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 235000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -4.7e-303:
		tmp = a * (1.0 / (k * k))
	elif k <= 235000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -4.7e-303)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (k <= 235000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -4.7e-303)
		tmp = a * (1.0 / (k * k));
	elseif (k <= 235000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -4.7e-303], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 235000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.6999999999999997e-303

   1. Initial program 85.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 85.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 28.2%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 29.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 29.3%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -4.6999999999999997e-303 < k < 235000

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + 10 \cdot k\right)}\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot 1\right) \cdot k\right)\right) \]

      2. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) \cdot k\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot \frac{1}{k}\right) \cdot {k}^{\color{blue}{2}}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) \cdot k\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(10 \cdot 1\right) \cdot k\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(10 \cdot k\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{10}\right)\right)\right) \]

      13. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{10}\right)\right)\right) \]

   10. Simplified 41.4%

       \[\leadsto \frac{a}{\color{blue}{1 + k \cdot 10}} \]

   if 235000 < k

   1. Initial program 82.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 82.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right) \]

      7. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right) \]

      9. +-lowering-+.f64 55.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right) \]

   10. Simplified 55.7%

       \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 235000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.074:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -4.7e-303)
   (* a (/ 1.0 (* k k)))
   (if (<= k 0.074) (+ a (* a (* k -10.0))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.7e-303) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 0.074) {
		tmp = a + (a * (k * -10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-4.7d-303)) then
        tmp = a * (1.0d0 / (k * k))
    else if (k <= 0.074d0) then
        tmp = a + (a * (k * (-10.0d0)))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.7e-303) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 0.074) {
		tmp = a + (a * (k * -10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -4.7e-303:
		tmp = a * (1.0 / (k * k))
	elif k <= 0.074:
		tmp = a + (a * (k * -10.0))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -4.7e-303)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (k <= 0.074)
		tmp = Float64(a + Float64(a * Float64(k * -10.0)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -4.7e-303)
		tmp = a * (1.0 / (k * k));
	elseif (k <= 0.074)
		tmp = a + (a * (k * -10.0));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -4.7e-303], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.074], N[(a + N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.6999999999999997e-303

   1. Initial program 85.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 85.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 28.2%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 29.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 29.3%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -4.6999999999999997e-303 < k < 0.0739999999999999963

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 42.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(-10 \cdot \left(a \cdot k\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot k\right) \cdot \color{blue}{-10}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \left(-10 \cdot \color{blue}{k}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

      7. *-lowering-*.f64 42.3%

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   10. Simplified 42.3%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]

   if 0.0739999999999999963 < k

   1. Initial program 83.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 83.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right) \]

      7. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right) \]

      9. +-lowering-+.f64 54.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right) \]

   10. Simplified 54.5%

       \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.074:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \frac{1}{k \cdot k}\\ \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (/ 1.0 (* k k)))))
   (if (<= k -4.7e-303) t_0 (if (<= k 0.1) (+ a (* a (* k -10.0))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * (1.0 / (k * k));
	double tmp;
	if (k <= -4.7e-303) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a + (a * (k * -10.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (1.0d0 / (k * k))
    if (k <= (-4.7d-303)) then
        tmp = t_0
    else if (k <= 0.1d0) then
        tmp = a + (a * (k * (-10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * (1.0 / (k * k));
	double tmp;
	if (k <= -4.7e-303) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a + (a * (k * -10.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * (1.0 / (k * k))
	tmp = 0
	if k <= -4.7e-303:
		tmp = t_0
	elif k <= 0.1:
		tmp = a + (a * (k * -10.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * Float64(1.0 / Float64(k * k)))
	tmp = 0.0
	if (k <= -4.7e-303)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = Float64(a + Float64(a * Float64(k * -10.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (1.0 / (k * k));
	tmp = 0.0;
	if (k <= -4.7e-303)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = a + (a * (k * -10.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.7e-303], t$95$0, If[LessEqual[k, 0.1], N[(a + N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -4.6999999999999997e-303 or 0.10000000000000001 < k

   1. Initial program 84.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 84.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 41.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 41.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 41.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 41.6%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -4.6999999999999997e-303 < k < 0.10000000000000001

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 42.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(-10 \cdot \left(a \cdot k\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot k\right) \cdot \color{blue}{-10}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \left(-10 \cdot \color{blue}{k}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

      7. *-lowering-*.f64 42.3%

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   10. Simplified 42.3%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \frac{1}{k \cdot k}\\ \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 235000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (/ 1.0 (* k k)))))
   (if (<= k -4.7e-303) t_0 (if (<= k 235000.0) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * (1.0 / (k * k));
	double tmp;
	if (k <= -4.7e-303) {
		tmp = t_0;
	} else if (k <= 235000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (1.0d0 / (k * k))
    if (k <= (-4.7d-303)) then
        tmp = t_0
    else if (k <= 235000.0d0) then
        tmp = a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * (1.0 / (k * k));
	double tmp;
	if (k <= -4.7e-303) {
		tmp = t_0;
	} else if (k <= 235000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * (1.0 / (k * k))
	tmp = 0
	if k <= -4.7e-303:
		tmp = t_0
	elif k <= 235000.0:
		tmp = a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * Float64(1.0 / Float64(k * k)))
	tmp = 0.0
	if (k <= -4.7e-303)
		tmp = t_0;
	elseif (k <= 235000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (1.0 / (k * k));
	tmp = 0.0;
	if (k <= -4.7e-303)
		tmp = t_0;
	elseif (k <= 235000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.7e-303], t$95$0, If[LessEqual[k, 235000.0], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -4.6999999999999997e-303 or 235000 < k

   1. Initial program 84.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 84.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 41.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 41.5%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{a}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{k \cdot k} \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{a}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), a\right) \]

       5. *-lowering-*.f64 42.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right) \]

   12. Applied egg-rr 42.0%

       \[\leadsto \color{blue}{\frac{1}{k \cdot k} \cdot a} \]

   if -4.6999999999999997e-303 < k < 235000

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 41.3%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 235000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -4.7 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 235000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -4.7e-303) t_0 (if (<= k 235000.0) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -4.7e-303) {
		tmp = t_0;
	} else if (k <= 235000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= (-4.7d-303)) then
        tmp = t_0
    else if (k <= 235000.0d0) then
        tmp = a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -4.7e-303) {
		tmp = t_0;
	} else if (k <= 235000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= -4.7e-303:
		tmp = t_0
	elif k <= 235000.0:
		tmp = a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -4.7e-303)
		tmp = t_0;
	elseif (k <= 235000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= -4.7e-303)
		tmp = t_0;
	elseif (k <= 235000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.7e-303], t$95$0, If[LessEqual[k, 235000.0], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -4.6999999999999997e-303 or 235000 < k

   1. Initial program 84.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 84.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 41.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 41.5%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -4.6999999999999997e-303 < k < 235000

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

      4. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

      20. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

   7. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 41.3%

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

Alternative 16: 19.9% accurate, 114.0× speedup

Math

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

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.5%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 89.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

3. Simplified 89.5%

   \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
4. Add Preprocessing

5. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot 10 + \color{blue}{k \cdot k}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot 1\right) + k \cdot k\right)\right)\right) \]

   4. lft-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right)\right) + k \cdot k\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) + k \cdot k\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k + \color{blue}{k} \cdot k\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right) + \color{blue}{k} \cdot k\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + k \cdot k\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{\color{blue}{2}}\right)\right)\right) \]

   10. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + 1 \cdot \color{blue}{{k}^{2}}\right)\right)\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

   19. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   20. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{\left(10 \cdot \frac{1}{k}\right)} \cdot k\right)\right)\right)\right) \]

   21. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right)\right)\right)\right) \]

   22. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10 \cdot 1\right)\right)\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + 10\right)\right)\right)\right) \]

7. Simplified 39.4%

   \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 16.7%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))