Falkner and Boettcher, Appendix A

Percentage Accurate: 89.7% → 97.5%

Time: 13.9s

Alternatives: 19

Speedup: 1.0×

• 22.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.9999999999999999e-154
  2. k = 0.0
  3. m = -2.00000000000000007e154

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: 89.7% 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: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.1e-7)
   (* (/ (pow k m) (+ (* k (+ k 10.0)) 1.0)) a)
   (/ (/ (* (pow k m) a) k) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1e-7) {
		tmp = (pow(k, m) / ((k * (k + 10.0)) + 1.0)) * a;
	} else {
		tmp = ((pow(k, m) * a) / 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 <= 2.1d-7) then
        tmp = ((k ** m) / ((k * (k + 10.0d0)) + 1.0d0)) * a
    else
        tmp = (((k ** m) * a) / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1e-7) {
		tmp = (Math.pow(k, m) / ((k * (k + 10.0)) + 1.0)) * a;
	} else {
		tmp = ((Math.pow(k, m) * a) / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.1e-7:
		tmp = (math.pow(k, m) / ((k * (k + 10.0)) + 1.0)) * a
	else:
		tmp = ((math.pow(k, m) * a) / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1e-7)
		tmp = Float64(Float64((k ^ m) / Float64(Float64(k * Float64(k + 10.0)) + 1.0)) * a);
	else
		tmp = Float64(Float64(Float64((k ^ m) * a) / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.1e-7)
		tmp = ((k ^ m) / ((k * (k + 10.0)) + 1.0)) * a;
	else
		tmp = (((k ^ m) * a) / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.1e-7], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.1e-7

   1. Initial program 99.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 99.4%

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

   3. Simplified 99.4%

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

      1. 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 99.4%

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

   6. Applied egg-rr 99.4%

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

   if 2.1e-7 < m

   1. Initial program 85.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 59.7%

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

   7. Simplified 59.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. pow-lowering-pow.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. pow-lowering-pow.f64 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;t\_0 \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 0.1) (* t_0 (+ (* k -10.0) 1.0)) (/ (/ t_0 k) k))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 0.1) {
		tmp = t_0 * ((k * -10.0) + 1.0);
	} else {
		tmp = (t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (k ** m) * a
    if (k <= 0.1d0) then
        tmp = t_0 * ((k * (-10.0d0)) + 1.0d0)
    else
        tmp = (t_0 / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if (k <= 0.1) {
		tmp = t_0 * ((k * -10.0) + 1.0);
	} else {
		tmp = (t_0 / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if k <= 0.1:
		tmp = t_0 * ((k * -10.0) + 1.0)
	else:
		tmp = (t_0 / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 0.1)
		tmp = Float64(t_0 * Float64(Float64(k * -10.0) + 1.0));
	else
		tmp = Float64(Float64(t_0 / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (k <= 0.1)
		tmp = t_0 * ((k * -10.0) + 1.0);
	else
		tmp = (t_0 / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 0.1], N[(t$95$0 * N[(N[(k * -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

         \[\leadsto -10 \cdot \left(\left(k \cdot {k}^{m}\right) \cdot a\right) + a \cdot {k}^{m} \]

      2. associate-*r* N/A

         \[\leadsto -10 \cdot \left(k \cdot \left({k}^{m} \cdot a\right)\right) + a \cdot {k}^{m} \]

      3. *-commutative N/A

         \[\leadsto -10 \cdot \left(k \cdot \left(a \cdot {k}^{m}\right)\right) + a \cdot {k}^{m} \]

      4. associate-*r* N/A

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

      5. *-commutative N/A

         \[\leadsto \left(k \cdot -10\right) \cdot \left(a \cdot {k}^{m}\right) + a \cdot {k}^{m} \]

      6. distribute-lft1-in N/A

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

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

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

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

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

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

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

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

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

      11. pow-lowering-pow.f64 100.0%

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

   7. Simplified 100.0%

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

   if 0.10000000000000001 < k

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.8%

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

   7. Simplified 86.8%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. pow-lowering-pow.f64 98.0%

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

   9. Applied egg-rr 98.0%

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. pow-lowering-pow.f64 98.1%

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

   11. Applied egg-rr 98.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -5.8e-8)
     t_0
     (if (<= m 1.85e-11)
       (/ -1.0 (- (/ -1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -5.8e-8) {
		tmp = t_0;
	} else if (m <= 1.85e-11) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / 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 = (k ** m) * a
    if (m <= (-5.8d-8)) then
        tmp = t_0
    else if (m <= 1.85d-11) then
        tmp = (-1.0d0) / (((-1.0d0) / a) - (k * ((k / a) + (10.0d0 / 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 = Math.pow(k, m) * a;
	double tmp;
	if (m <= -5.8e-8) {
		tmp = t_0;
	} else if (m <= 1.85e-11) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if m <= -5.8e-8:
		tmp = t_0
	elif m <= 1.85e-11:
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -5.8e-8)
		tmp = t_0;
	elseif (m <= 1.85e-11)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / a) - Float64(k * Float64(Float64(k / a) + Float64(10.0 / a)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (m <= -5.8e-8)
		tmp = t_0;
	elseif (m <= 1.85e-11)
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -5.8e-8], t$95$0, If[LessEqual[m, 1.85e-11], N[(-1.0 / N[(N[(-1.0 / a), $MachinePrecision] - N[(k * N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.8000000000000003e-8 or 1.8500000000000001e-11 < m

   1. Initial program 93.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 98.9%

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

   7. Simplified 98.9%

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

   if -5.8000000000000003e-8 < m < 1.8500000000000001e-11

   1. Initial program 98.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.2%

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

   7. Simplified 98.2%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

       5. +-commutative N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. /-lowering-/.f64 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a))) (if (<= k 0.105) t_0 (/ (/ t_0 k) k))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 0.105) {
		tmp = t_0;
	} else {
		tmp = (t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (k ** m) * a
    if (k <= 0.105d0) then
        tmp = t_0
    else
        tmp = (t_0 / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if (k <= 0.105) {
		tmp = t_0;
	} else {
		tmp = (t_0 / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if k <= 0.105:
		tmp = t_0
	else:
		tmp = (t_0 / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 0.105)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_0 / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (k <= 0.105)
		tmp = t_0;
	else
		tmp = (t_0 / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 0.105], t$95$0, N[(N[(t$95$0 / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.104999999999999996

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 99.7%

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

   7. Simplified 99.7%

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

   if 0.104999999999999996 < k

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.8%

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

   7. Simplified 86.8%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. pow-lowering-pow.f64 98.0%

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

   9. Applied egg-rr 98.0%

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. pow-lowering-pow.f64 98.1%

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

   11. Applied egg-rr 98.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.105) (* (pow k m) a) (/ (* a (/ (pow k m) k)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.105) {
		tmp = pow(k, m) * a;
	} else {
		tmp = (a * (pow(k, m) / 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 (k <= 0.105d0) then
        tmp = (k ** m) * a
    else
        tmp = (a * ((k ** m) / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.105) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = (a * (Math.pow(k, m) / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.105:
		tmp = math.pow(k, m) * a
	else:
		tmp = (a * (math.pow(k, m) / k)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.105)
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(Float64(a * Float64((k ^ m) / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.105)
		tmp = (k ^ m) * a;
	else
		tmp = (a * ((k ^ m) / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.105], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(N[(a * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.104999999999999996

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 99.7%

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

   7. Simplified 99.7%

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

   if 0.104999999999999996 < k

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.8%

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

   7. Simplified 86.8%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. pow-lowering-pow.f64 98.0%

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

   9. Applied egg-rr 98.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.105) (* (pow k m) a) (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.105) {
		tmp = pow(k, m) * a;
	} 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 <= 0.105d0) then
        tmp = (k ** m) * a
    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 <= 0.105) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.105:
		tmp = math.pow(k, m) * a
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.105)
		tmp = Float64((k ^ m) * a);
	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 <= 0.105)
		tmp = (k ^ m) * a;
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.105], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.104999999999999996

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 99.7%

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

   7. Simplified 99.7%

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

   if 0.104999999999999996 < k

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.8%

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

   7. Simplified 86.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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 96.8%

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

   9. Applied egg-rr 96.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.105:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.2:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.8:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{\frac{\frac{100 + \frac{-1000}{k}}{k} - 10}{k} + 1}{k \cdot k} \cdot \left(k \cdot 10 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.2)
   (/ (/ (/ (* a 99.0) k) k) (* k k))
   (if (<= m 0.8)
     (/ -1.0 (- (/ -1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
     (/
      a
      (*
       (/ (+ (/ (- (/ (+ 100.0 (/ -1000.0 k)) k) 10.0) k) 1.0) (* k k))
       (+ (* k 10.0) 1.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.2) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 0.8) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a / (((((((100.0 + (-1000.0 / k)) / k) - 10.0) / k) + 1.0) / (k * k)) * ((k * 10.0) + 1.0));
	}
	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 <= (-0.2d0)) then
        tmp = (((a * 99.0d0) / k) / k) / (k * k)
    else if (m <= 0.8d0) then
        tmp = (-1.0d0) / (((-1.0d0) / a) - (k * ((k / a) + (10.0d0 / a))))
    else
        tmp = a / (((((((100.0d0 + ((-1000.0d0) / k)) / k) - 10.0d0) / k) + 1.0d0) / (k * k)) * ((k * 10.0d0) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.2) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 0.8) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a / (((((((100.0 + (-1000.0 / k)) / k) - 10.0) / k) + 1.0) / (k * k)) * ((k * 10.0) + 1.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.2:
		tmp = (((a * 99.0) / k) / k) / (k * k)
	elif m <= 0.8:
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))))
	else:
		tmp = a / (((((((100.0 + (-1000.0 / k)) / k) - 10.0) / k) + 1.0) / (k * k)) * ((k * 10.0) + 1.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.2)
		tmp = Float64(Float64(Float64(Float64(a * 99.0) / k) / k) / Float64(k * k));
	elseif (m <= 0.8)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / a) - Float64(k * Float64(Float64(k / a) + Float64(10.0 / a)))));
	else
		tmp = Float64(a / Float64(Float64(Float64(Float64(Float64(Float64(Float64(100.0 + Float64(-1000.0 / k)) / k) - 10.0) / k) + 1.0) / Float64(k * k)) * Float64(Float64(k * 10.0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.2)
		tmp = (((a * 99.0) / k) / k) / (k * k);
	elseif (m <= 0.8)
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	else
		tmp = a / (((((((100.0 + (-1000.0 / k)) / k) - 10.0) / k) + 1.0) / (k * k)) * ((k * 10.0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.2], N[(N[(N[(N[(a * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.8], N[(-1.0 / N[(N[(-1.0 / a), $MachinePrecision] - N[(k * N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(N[(N[(N[(N[(N[(100.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] - 10.0), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.20000000000000001

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 67.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(99 \cdot \frac{a}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}}{k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{99 \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot \frac{a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot \frac{a}{k}\right), k\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{99 \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(-99 \cdot a\right)\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       17. *-lowering-*.f64 76.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{a \cdot 99}{k}}{k}}}{k \cdot k} \]

   if -0.20000000000000001 < m < 0.80000000000000004

   1. Initial program 98.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 95.3%

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

   7. Simplified 95.3%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 95.2%

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

   9. Applied egg-rr 95.2%

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

   10. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

       5. +-commutative N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. /-lowering-/.f64 96.3%

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

   12. Simplified 96.3%

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

   if 0.80000000000000004 < m

   1. Initial program 85.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 85.0%

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

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 10.3%

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

   10. Taylor expanded in k around -inf

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

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

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

   12. Simplified 48.7%

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

   13. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 59.9%

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

   15. Simplified 59.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.2:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.8:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{\frac{\frac{100 + \frac{-1000}{k}}{k} - 10}{k} + 1}{k \cdot k} \cdot \left(k \cdot 10 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -0.17:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{1}{k \cdot k} \cdot \left(t\_0 \cdot \left(t\_0 + 1\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -0.17)
     (/ (/ (/ (* a 99.0) k) k) (* k k))
     (if (<= m 2.45e+32)
       (/ -1.0 (- (/ -1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
       (/ a (* (/ 1.0 (* k k)) (+ (* t_0 (+ t_0 1.0)) 1.0)))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.17) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 2.45e+32) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a / ((1.0 / (k * k)) * ((t_0 * (t_0 + 1.0)) + 1.0));
	}
	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 = k * (k + 10.0d0)
    if (m <= (-0.17d0)) then
        tmp = (((a * 99.0d0) / k) / k) / (k * k)
    else if (m <= 2.45d+32) then
        tmp = (-1.0d0) / (((-1.0d0) / a) - (k * ((k / a) + (10.0d0 / a))))
    else
        tmp = a / ((1.0d0 / (k * k)) * ((t_0 * (t_0 + 1.0d0)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.17) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 2.45e+32) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a / ((1.0 / (k * k)) * ((t_0 * (t_0 + 1.0)) + 1.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.17:
		tmp = (((a * 99.0) / k) / k) / (k * k)
	elif m <= 2.45e+32:
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))))
	else:
		tmp = a / ((1.0 / (k * k)) * ((t_0 * (t_0 + 1.0)) + 1.0))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.17)
		tmp = Float64(Float64(Float64(Float64(a * 99.0) / k) / k) / Float64(k * k));
	elseif (m <= 2.45e+32)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / a) - Float64(k * Float64(Float64(k / a) + Float64(10.0 / a)))));
	else
		tmp = Float64(a / Float64(Float64(1.0 / Float64(k * k)) * Float64(Float64(t_0 * Float64(t_0 + 1.0)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.17)
		tmp = (((a * 99.0) / k) / k) / (k * k);
	elseif (m <= 2.45e+32)
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	else
		tmp = a / ((1.0 / (k * k)) * ((t_0 * (t_0 + 1.0)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.17], N[(N[(N[(N[(a * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.45e+32], N[(-1.0 / N[(N[(-1.0 / a), $MachinePrecision] - N[(k * N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.170000000000000012

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 67.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(99 \cdot \frac{a}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}}{k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{99 \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot \frac{a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot \frac{a}{k}\right), k\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{99 \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(-99 \cdot a\right)\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       17. *-lowering-*.f64 76.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{a \cdot 99}{k}}{k}}}{k \cdot k} \]

   if -0.170000000000000012 < m < 2.4500000000000001e32

   1. Initial program 95.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.7%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 87.4%

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

   9. Applied egg-rr 87.4%

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

   10. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

       5. +-commutative N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. /-lowering-/.f64 88.4%

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

   12. Simplified 88.4%

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

   if 2.4500000000000001e32 < m

   1. Initial program 87.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.5%

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

   3. Simplified 87.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 3.6%

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

   7. Simplified 3.6%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 11.3%

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

   10. Taylor expanded in k around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 40.7%

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

   12. Simplified 40.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.17:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{1}{k \cdot k} \cdot \left(\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right) + 1\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.64:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 23.5:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right) + a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.64)
   (/ (/ (/ (* a 99.0) k) k) (* k k))
   (if (<= m 23.5)
     (/ -1.0 (- (/ -1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
     (+ a (* k (+ (* k (* a 99.0)) (* a -10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.64) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 23.5) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a + (k * ((k * (a * 99.0)) + (a * -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 <= (-0.64d0)) then
        tmp = (((a * 99.0d0) / k) / k) / (k * k)
    else if (m <= 23.5d0) then
        tmp = (-1.0d0) / (((-1.0d0) / a) - (k * ((k / a) + (10.0d0 / a))))
    else
        tmp = a + (k * ((k * (a * 99.0d0)) + (a * (-10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.64) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 23.5) {
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a + (k * ((k * (a * 99.0)) + (a * -10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.64:
		tmp = (((a * 99.0) / k) / k) / (k * k)
	elif m <= 23.5:
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))))
	else:
		tmp = a + (k * ((k * (a * 99.0)) + (a * -10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.64)
		tmp = Float64(Float64(Float64(Float64(a * 99.0) / k) / k) / Float64(k * k));
	elseif (m <= 23.5)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / a) - Float64(k * Float64(Float64(k / a) + Float64(10.0 / a)))));
	else
		tmp = Float64(a + Float64(k * Float64(Float64(k * Float64(a * 99.0)) + Float64(a * -10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.64)
		tmp = (((a * 99.0) / k) / k) / (k * k);
	elseif (m <= 23.5)
		tmp = -1.0 / ((-1.0 / a) - (k * ((k / a) + (10.0 / a))));
	else
		tmp = a + (k * ((k * (a * 99.0)) + (a * -10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.64], N[(N[(N[(N[(a * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 23.5], N[(-1.0 / N[(N[(-1.0 / a), $MachinePrecision] - N[(k * N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision] + N[(a * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.640000000000000013

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 67.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(99 \cdot \frac{a}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}}{k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{99 \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot \frac{a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot \frac{a}{k}\right), k\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{99 \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(-99 \cdot a\right)\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       17. *-lowering-*.f64 76.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{a \cdot 99}{k}}{k}}}{k \cdot k} \]

   if -0.640000000000000013 < m < 23.5

   1. Initial program 98.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 94.3%

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

   7. Simplified 94.3%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 94.2%

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

   9. Applied egg-rr 94.2%

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

   10. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

       5. +-commutative N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. /-lowering-/.f64 95.3%

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

   12. Simplified 95.3%

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

   if 23.5 < m

   1. Initial program 84.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 84.8%

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

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. 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. +-lowering-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right)\right), \left(-10 \cdot a\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(99 \cdot a\right)\right), \left(-10 \cdot a\right)\right)\right)\right) \]

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(99, a\right)\right), \left(-10 \cdot a\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(99, a\right)\right), \left(a \cdot \color{blue}{-10}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(99, a\right)\right), \mathsf{*.f64}\left(a, \color{blue}{-10}\right)\right)\right)\right) \]

   10. Simplified 23.9%

       \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(99 \cdot a\right) + a \cdot -10\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.64:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 23.5:\\ \;\;\;\;\frac{-1}{\frac{-1}{a} - k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right) + a \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.175:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 23.5:\\ \;\;\;\;a \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right) + a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.175)
   (/ (/ (/ (* a 99.0) k) k) (* k k))
   (if (<= m 23.5)
     (* a (/ -1.0 (- -1.0 (* k (+ k 10.0)))))
     (+ a (* k (+ (* k (* a 99.0)) (* a -10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.175) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 23.5) {
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = a + (k * ((k * (a * 99.0)) + (a * -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 <= (-0.175d0)) then
        tmp = (((a * 99.0d0) / k) / k) / (k * k)
    else if (m <= 23.5d0) then
        tmp = a * ((-1.0d0) / ((-1.0d0) - (k * (k + 10.0d0))))
    else
        tmp = a + (k * ((k * (a * 99.0d0)) + (a * (-10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.175) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else if (m <= 23.5) {
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = a + (k * ((k * (a * 99.0)) + (a * -10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.175:
		tmp = (((a * 99.0) / k) / k) / (k * k)
	elif m <= 23.5:
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))))
	else:
		tmp = a + (k * ((k * (a * 99.0)) + (a * -10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.175)
		tmp = Float64(Float64(Float64(Float64(a * 99.0) / k) / k) / Float64(k * k));
	elseif (m <= 23.5)
		tmp = Float64(a * Float64(-1.0 / Float64(-1.0 - Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a + Float64(k * Float64(Float64(k * Float64(a * 99.0)) + Float64(a * -10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.175)
		tmp = (((a * 99.0) / k) / k) / (k * k);
	elseif (m <= 23.5)
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	else
		tmp = a + (k * ((k * (a * 99.0)) + (a * -10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.175], N[(N[(N[(N[(a * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 23.5], N[(a * N[(-1.0 / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision] + N[(a * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.17499999999999999

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 67.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(99 \cdot \frac{a}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}}{k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{99 \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot \frac{a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot \frac{a}{k}\right), k\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{99 \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(-99 \cdot a\right)\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       17. *-lowering-*.f64 76.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{a \cdot 99}{k}}{k}}}{k \cdot k} \]

   if -0.17499999999999999 < m < 23.5

   1. Initial program 98.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 94.3%

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

   7. Simplified 94.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 23.5 < m

   1. Initial program 84.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 84.8%

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

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. 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. +-lowering-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right)\right), \left(-10 \cdot a\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(99 \cdot a\right)\right), \left(-10 \cdot a\right)\right)\right)\right) \]

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(99, a\right)\right), \left(-10 \cdot a\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(99, a\right)\right), \left(a \cdot \color{blue}{-10}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(99, a\right)\right), \mathsf{*.f64}\left(a, \color{blue}{-10}\right)\right)\right)\right) \]

   10. Simplified 23.9%

       \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(99 \cdot a\right) + a \cdot -10\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.175:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 23.5:\\ \;\;\;\;a \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right) + a \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -4.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 550000:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -4.8e-283)
   (/ a (* k k))
   (if (<= k 550000.0) (/ a (+ (* k 10.0) 1.0)) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.8e-283) {
		tmp = a / (k * k);
	} else if (k <= 550000.0) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = (a / 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 (k <= (-4.8d-283)) then
        tmp = a / (k * k)
    else if (k <= 550000.0d0) then
        tmp = a / ((k * 10.0d0) + 1.0d0)
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.8e-283) {
		tmp = a / (k * k);
	} else if (k <= 550000.0) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -4.8e-283:
		tmp = a / (k * k)
	elif k <= 550000.0:
		tmp = a / ((k * 10.0) + 1.0)
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -4.8e-283)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 550000.0)
		tmp = Float64(a / Float64(Float64(k * 10.0) + 1.0));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -4.8e-283)
		tmp = a / (k * k);
	elseif (k <= 550000.0)
		tmp = a / ((k * 10.0) + 1.0);
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -4.8e-283], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 550000.0], N[(a / N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.7999999999999999e-283

   1. Initial program 95.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.7%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 12.0%

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

   7. Simplified 12.0%

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

   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 29.9%

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

   10. Simplified 29.9%

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

   if -4.7999999999999999e-283 < k < 5.5e5

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 45.0%

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

   7. Simplified 45.0%

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

   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. +-lowering-+.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 44.2%

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

   10. Simplified 44.2%

       \[\leadsto \frac{a}{\color{blue}{1 + k \cdot 10}} \]

   if 5.5e5 < k

   1. Initial program 87.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.9%

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

   3. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 65.4%

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

   7. Simplified 65.4%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 65.4%

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

   9. Applied egg-rr 65.4%

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

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

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

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

       4. /-lowering-/.f64 64.9%

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

   12. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 550000:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -4.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.106:\\ \;\;\;\;a + a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -4.8e-283)
   (/ a (* k k))
   (if (<= k 0.106) (+ a (* a (* k -10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.8e-283) {
		tmp = a / (k * k);
	} else if (k <= 0.106) {
		tmp = a + (a * (k * -10.0));
	} else {
		tmp = (a / 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 (k <= (-4.8d-283)) then
        tmp = a / (k * k)
    else if (k <= 0.106d0) then
        tmp = a + (a * (k * (-10.0d0)))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.8e-283) {
		tmp = a / (k * k);
	} else if (k <= 0.106) {
		tmp = a + (a * (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -4.8e-283:
		tmp = a / (k * k)
	elif k <= 0.106:
		tmp = a + (a * (k * -10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -4.8e-283)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.106)
		tmp = Float64(a + Float64(a * Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -4.8e-283)
		tmp = a / (k * k);
	elseif (k <= 0.106)
		tmp = a + (a * (k * -10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -4.8e-283], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.106], N[(a + N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.7999999999999999e-283

   1. Initial program 95.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.7%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 12.0%

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

   7. Simplified 12.0%

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

   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 29.9%

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

   10. Simplified 29.9%

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

   if -4.7999999999999999e-283 < k < 0.105999999999999997

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in k around 0

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 45.7%

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

   10. Simplified 45.7%

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

   if 0.105999999999999997 < k

   1. Initial program 88.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.5%

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

   3. Simplified 88.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 63.7%

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

   7. Simplified 63.7%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 63.6%

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

   9. Applied egg-rr 63.6%

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

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

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

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

       4. /-lowering-/.f64 62.2%

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

   12. Simplified 62.2%

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

Alternative 13: 56.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.39:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.39)
   (/ (/ (/ (* a 99.0) k) k) (* k k))
   (* a (/ -1.0 (- -1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.39) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else {
		tmp = a * (-1.0 / (-1.0 - (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 (m <= (-0.39d0)) then
        tmp = (((a * 99.0d0) / k) / k) / (k * k)
    else
        tmp = a * ((-1.0d0) / ((-1.0d0) - (k * (k + 10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.39) {
		tmp = (((a * 99.0) / k) / k) / (k * k);
	} else {
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.39:
		tmp = (((a * 99.0) / k) / k) / (k * k)
	else:
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.39)
		tmp = Float64(Float64(Float64(Float64(a * 99.0) / k) / k) / Float64(k * k));
	else
		tmp = Float64(a * Float64(-1.0 / Float64(-1.0 - Float64(k * Float64(k + 10.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.39)
		tmp = (((a * 99.0) / k) / k) / (k * k);
	else
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.39], N[(N[(N[(N[(a * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(-1.0 / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.39000000000000001

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 67.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(99 \cdot \frac{a}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}}{k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{99 \cdot a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{99 \cdot \frac{a}{k}}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot \frac{a}{k}\right), k\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{99 \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-99 \cdot a\right)}{k}\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(-99 \cdot a\right)\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-99\right)\right) \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(99 \cdot a\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       17. *-lowering-*.f64 76.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 99\right), k\right), k\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{a \cdot 99}{k}}{k}}}{k \cdot k} \]

   if -0.39000000000000001 < m

   1. Initial program 92.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 92.2%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 51.3%

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

   7. Simplified 51.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 51.3%

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

   9. Applied egg-rr 51.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.39:\\ \;\;\;\;\frac{\frac{\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.64:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.64) (/ a (* k k)) (* a (/ -1.0 (- -1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.64) {
		tmp = a / (k * k);
	} else {
		tmp = a * (-1.0 / (-1.0 - (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 (m <= (-0.64d0)) then
        tmp = a / (k * k)
    else
        tmp = a * ((-1.0d0) / ((-1.0d0) - (k * (k + 10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.64) {
		tmp = a / (k * k);
	} else {
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.64:
		tmp = a / (k * k)
	else:
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.64)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a * Float64(-1.0 / Float64(-1.0 - Float64(k * Float64(k + 10.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.64)
		tmp = a / (k * k);
	else
		tmp = a * (-1.0 / (-1.0 - (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.64], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(-1.0 / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.640000000000000013

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   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 62.5%

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

   10. Simplified 62.5%

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

   if -0.640000000000000013 < m

   1. Initial program 92.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 92.2%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 51.3%

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

   7. Simplified 51.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 51.3%

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

   9. Applied egg-rr 51.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.64:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -4.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -4.8e-283) (/ a (* k k)) (if (<= k 1.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.8e-283) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / 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 (k <= (-4.8d-283)) then
        tmp = a / (k * k)
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -4.8e-283) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -4.8e-283:
		tmp = a / (k * k)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -4.8e-283)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -4.8e-283)
		tmp = a / (k * k);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -4.8e-283], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.7999999999999999e-283

   1. Initial program 95.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.7%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 12.0%

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

   7. Simplified 12.0%

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

   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 29.9%

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

   10. Simplified 29.9%

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

   if -4.7999999999999999e-283 < k < 1

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 45.3%

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

   if 1 < k

   1. Initial program 88.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.5%

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

   3. Simplified 88.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 63.7%

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

   7. Simplified 63.7%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 63.6%

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

   9. Applied egg-rr 63.6%

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

   10. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

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

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

       4. /-lowering-/.f64 62.2%

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

   12. Simplified 62.2%

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

Alternative 16: 45.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -4.8 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))) (if (<= k -4.8e-283) t_0 (if (<= k 1.0) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -4.8e-283) {
		tmp = t_0;
	} else if (k <= 1.0) {
		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 <= (-4.8d-283)) then
        tmp = t_0
    else if (k <= 1.0d0) 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 <= -4.8e-283) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= -4.8e-283:
		tmp = t_0
	elif k <= 1.0:
		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 <= -4.8e-283)
		tmp = t_0;
	elseif (k <= 1.0)
		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 <= -4.8e-283)
		tmp = t_0;
	elseif (k <= 1.0)
		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, -4.8e-283], t$95$0, If[LessEqual[k, 1.0], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -4.7999999999999999e-283 or 1 < k

   1. Initial program 91.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.6%

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

   3. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 40.8%

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

   7. Simplified 40.8%

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

   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 47.7%

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

   10. Simplified 47.7%

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

   if -4.7999999999999999e-283 < k < 1

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 45.3%

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

Alternative 17: 52.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.045:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.045) (/ a (* k k)) (/ a (+ (* k (+ k 10.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.045) {
		tmp = a / (k * k);
	} else {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	}
	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 <= (-0.045d0)) then
        tmp = a / (k * k)
    else
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.045) {
		tmp = a / (k * k);
	} else {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.045:
		tmp = a / (k * k)
	else:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.045)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.045)
		tmp = a / (k * k);
	else
		tmp = a / ((k * (k + 10.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.045], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.044999999999999998

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   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 62.5%

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

   10. Simplified 62.5%

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

   if -0.044999999999999998 < m

   1. Initial program 92.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 92.2%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 51.3%

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

   7. Simplified 51.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.045:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.7% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.26:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.26) (/ a (* k k)) (/ a (+ (* k k) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.26) {
		tmp = a / (k * k);
	} else {
		tmp = a / ((k * k) + 1.0);
	}
	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 <= (-0.26d0)) then
        tmp = a / (k * k)
    else
        tmp = a / ((k * k) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.26) {
		tmp = a / (k * k);
	} else {
		tmp = a / ((k * k) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.26:
		tmp = a / (k * k)
	else:
		tmp = a / ((k * k) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.26)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(Float64(k * k) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.26)
		tmp = a / (k * k);
	else
		tmp = a / ((k * k) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.26], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.26000000000000001

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 26.6%

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

   7. Simplified 26.6%

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

   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 62.5%

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

   10. Simplified 62.5%

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

   if -0.26000000000000001 < m

   1. Initial program 92.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 92.2%

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

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 51.3%

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

   7. Simplified 51.3%

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

   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 50.1%

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

   10. Simplified 50.1%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.26:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 20.2% 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 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. +-lowering-+.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 42.7%

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

7. Simplified 42.7%

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

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 20.2%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))