Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 98.9%

Time: 11.8s

Alternatives: 13

Speedup: 1.0×

• 21.7% 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: 89.9% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3e-5)
   (* a (pow k m))
   (if (<= m 1.3e-13)
     (/ 1.0 (+ (/ 1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
     (* a (pow k (- m 2.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3e-5) {
		tmp = a * pow(k, m);
	} else if (m <= 1.3e-13) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / 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 (m <= (-3d-5)) then
        tmp = a * (k ** m)
    else if (m <= 1.3d-13) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((k / a) + (10.0d0 / 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 (m <= -3e-5) {
		tmp = a * Math.pow(k, m);
	} else if (m <= 1.3e-13) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3e-5:
		tmp = a * math.pow(k, m)
	elif m <= 1.3e-13:
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))))
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3e-5)
		tmp = Float64(a * (k ^ m));
	elseif (m <= 1.3e-13)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(k / a) + Float64(10.0 / 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 (m <= -3e-5)
		tmp = a * (k ^ m);
	elseif (m <= 1.3e-13)
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3e-5], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3e-13], 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[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.00000000000000008e-5

   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 k around 0

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

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

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

      2. pow-lowering-pow.f64 100.0%

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

   7. Simplified 100.0%

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

   if -3.00000000000000008e-5 < m < 1.3e-13

   1. Initial program 95.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 94.8%

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

   7. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   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 94.0%

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

   9. Applied egg-rr 94.0%

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

   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 98.3%

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

   12. Simplified 98.3%

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

   if 1.3e-13 < m

   1. Initial program 75.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in k around 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 48.7%

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

   7. Simplified 48.7%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.0285:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -0.002)
     t_0
     (if (<= m 0.0285)
       (/ 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 = a * pow(k, m);
	double tmp;
	if (m <= -0.002) {
		tmp = t_0;
	} else if (m <= 0.0285) {
		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 = a * (k ** m)
    if (m <= (-0.002d0)) then
        tmp = t_0
    else if (m <= 0.0285d0) 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 = a * Math.pow(k, m);
	double tmp;
	if (m <= -0.002) {
		tmp = t_0;
	} else if (m <= 0.0285) {
		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 = a * math.pow(k, m)
	tmp = 0
	if m <= -0.002:
		tmp = t_0
	elif m <= 0.0285:
		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(a * (k ^ m))
	tmp = 0.0
	if (m <= -0.002)
		tmp = t_0;
	elseif (m <= 0.0285)
		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 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -0.002)
		tmp = t_0;
	elseif (m <= 0.0285)
		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[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.002], t$95$0, If[LessEqual[m, 0.0285], 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 < -2e-3 or 0.028500000000000001 < m

   1. Initial program 87.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.2%

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

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 100.0%

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

   7. Simplified 100.0%

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

   if -2e-3 < m < 0.028500000000000001

   1. Initial program 95.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 94.6%

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

   7. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   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 93.8%

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

   9. Applied egg-rr 93.8%

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

   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 98.1%

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

   12. Simplified 98.1%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 0.24:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= k 0.24) t_0 (/ (/ t_0 k) k))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 0.24) {
		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 = a * (k ** m)
    if (k <= 0.24d0) 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 = a * Math.pow(k, m);
	double tmp;
	if (k <= 0.24) {
		tmp = t_0;
	} else {
		tmp = (t_0 / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= 0.24:
		tmp = t_0
	else:
		tmp = (t_0 / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 0.24)
		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 = a * (k ^ m);
	tmp = 0.0;
	if (k <= 0.24)
		tmp = t_0;
	else
		tmp = (t_0 / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.24], t$95$0, N[(N[(t$95$0 / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.23999999999999999

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.2%

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

   3. Simplified 96.2%

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

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

   7. Simplified 99.4%

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

   if 0.23999999999999999 < k

   1. Initial program 81.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   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 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. pow-lowering-pow.f64 97.4%

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

   10. Simplified 97.4%

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

Alternative 4: 75.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a - \frac{a \cdot 10 + \frac{a \cdot -99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.67:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.5e+18)
   (/ (- a (/ (+ (* a 10.0) (/ (* a -99.0) k)) k)) (* k k))
   (if (<= m 0.67)
     (/ 1.0 (+ (/ 1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
     (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = (a - (((a * 10.0) + ((a * -99.0) / k)) / k)) / (k * k);
	} else if (m <= 0.67) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.5d+18)) then
        tmp = (a - (((a * 10.0d0) + ((a * (-99.0d0)) / k)) / k)) / (k * k)
    else if (m <= 0.67d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((k / a) + (10.0d0 / a))))
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = (a - (((a * 10.0) + ((a * -99.0) / k)) / k)) / (k * k);
	} else if (m <= 0.67) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.5e+18:
		tmp = (a - (((a * 10.0) + ((a * -99.0) / k)) / k)) / (k * k)
	elif m <= 0.67:
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))))
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.5e+18)
		tmp = Float64(Float64(a - Float64(Float64(Float64(a * 10.0) + Float64(Float64(a * -99.0) / k)) / k)) / Float64(k * k));
	elseif (m <= 0.67)
		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(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.5e+18)
		tmp = (a - (((a * 10.0) + ((a * -99.0) / k)) / k)) / (k * k);
	elseif (m <= 0.67)
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.5e+18], N[(N[(a - N[(N[(N[(a * 10.0), $MachinePrecision] + N[(N[(a * -99.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.67], 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[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.5e18

   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 36.6%

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

   7. Simplified 36.6%

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

   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 63.0%

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

   if -4.5e18 < m < 0.67000000000000004

   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 92.3%

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

   7. Simplified 92.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   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 91.5%

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

   9. Applied egg-rr 91.5%

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

   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 95.7%

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

   12. Simplified 95.7%

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

   if 0.67000000000000004 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.12:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.5e+18)
   (/ a (* k k))
   (if (<= m 1.12)
     (/ 1.0 (+ (/ 1.0 a) (* k (+ (/ k a) (/ 10.0 a)))))
     (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 1.12) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.5d+18)) then
        tmp = a / (k * k)
    else if (m <= 1.12d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((k / a) + (10.0d0 / a))))
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 1.12) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.5e+18:
		tmp = a / (k * k)
	elif m <= 1.12:
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))))
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.5e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.12)
		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(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.5e+18)
		tmp = a / (k * k);
	elseif (m <= 1.12)
		tmp = 1.0 / ((1.0 / a) + (k * ((k / a) + (10.0 / a))));
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.5e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.12], 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[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.5e18

   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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 60.1%

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

   if -4.5e18 < m < 1.1200000000000001

   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 92.3%

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

   7. Simplified 92.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   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 91.5%

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

   9. Applied egg-rr 91.5%

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

   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 95.7%

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

   12. Simplified 95.7%

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

   if 1.1200000000000001 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.85 \cdot 10^{-25}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.236:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.85e-25)
   (/ a (* k k))
   (if (<= m 1.32e-101)
     (/ (/ a k) k)
     (if (<= m 0.236) a (* a (* (* k k) 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.85e-25) {
		tmp = a / (k * k);
	} else if (m <= 1.32e-101) {
		tmp = (a / k) / k;
	} else if (m <= 0.236) {
		tmp = a;
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.85d-25)) then
        tmp = a / (k * k)
    else if (m <= 1.32d-101) then
        tmp = (a / k) / k
    else if (m <= 0.236d0) then
        tmp = a
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.85e-25) {
		tmp = a / (k * k);
	} else if (m <= 1.32e-101) {
		tmp = (a / k) / k;
	} else if (m <= 0.236) {
		tmp = a;
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.85e-25:
		tmp = a / (k * k)
	elif m <= 1.32e-101:
		tmp = (a / k) / k
	elif m <= 0.236:
		tmp = a
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.85e-25)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.32e-101)
		tmp = Float64(Float64(a / k) / k);
	elseif (m <= 0.236)
		tmp = a;
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.85e-25)
		tmp = a / (k * k);
	elseif (m <= 1.32e-101)
		tmp = (a / k) / k;
	elseif (m <= 0.236)
		tmp = a;
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.85e-25], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.32e-101], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.236], a, N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -1.85000000000000004e-25

   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 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 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 59.4%

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

   if -1.85000000000000004e-25 < m < 1.32e-101

   1. Initial program 94.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in k around 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 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. pow-lowering-pow.f64 55.5%

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

   10. Simplified 55.5%

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

   11. Taylor expanded in m around 0

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

       1. /-lowering-/.f64 55.5%

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

   13. Simplified 55.5%

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

   if 1.32e-101 < m < 0.23599999999999999

   1. Initial program 99.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 68.4%

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

   if 0.23599999999999999 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.98:\\ \;\;\;\;\frac{a}{1 + \frac{k}{\frac{1}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.5e+18)
   (/ a (* k k))
   (if (<= m 0.98)
     (/ a (+ 1.0 (/ k (/ 1.0 (+ k 10.0)))))
     (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 0.98) {
		tmp = a / (1.0 + (k / (1.0 / (k + 10.0))));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.5d+18)) then
        tmp = a / (k * k)
    else if (m <= 0.98d0) then
        tmp = a / (1.0d0 + (k / (1.0d0 / (k + 10.0d0))))
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 0.98) {
		tmp = a / (1.0 + (k / (1.0 / (k + 10.0))));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.5e+18:
		tmp = a / (k * k)
	elif m <= 0.98:
		tmp = a / (1.0 + (k / (1.0 / (k + 10.0))))
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.5e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.98)
		tmp = Float64(a / Float64(1.0 + Float64(k / Float64(1.0 / Float64(k + 10.0)))));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.5e+18)
		tmp = a / (k * k);
	elseif (m <= 0.98)
		tmp = a / (1.0 + (k / (1.0 / (k + 10.0))));
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.5e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.98], N[(a / N[(1.0 + N[(k / N[(1.0 / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.5e18

   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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 60.1%

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

   if -4.5e18 < m < 0.97999999999999998

   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 92.3%

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

   7. Simplified 92.3%

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

      1. /-rgt-identity N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-lowering-+.f64 92.3%

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

   9. Applied egg-rr 92.3%

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

   if 0.97999999999999998 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.5e+18)
   (/ a (* k k))
   (if (<= m 1.3) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 1.3) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.5d+18)) then
        tmp = a / (k * k)
    else if (m <= 1.3d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 1.3) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.5e+18:
		tmp = a / (k * k)
	elif m <= 1.3:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.5e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.3)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.5e+18)
		tmp = a / (k * k);
	elseif (m <= 1.3)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.5e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.5e18

   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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 60.1%

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

   if -4.5e18 < m < 1.30000000000000004

   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 92.3%

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

   7. Simplified 92.3%

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

   if 1.30000000000000004 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.84:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.5e+18)
   (/ a (* k k))
   (if (<= m 0.84) (/ a (+ 1.0 (* k k))) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 0.84) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.5d+18)) then
        tmp = a / (k * k)
    else if (m <= 0.84d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e+18) {
		tmp = a / (k * k);
	} else if (m <= 0.84) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.5e+18:
		tmp = a / (k * k)
	elif m <= 0.84:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.5e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.84)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.5e+18)
		tmp = a / (k * k);
	elseif (m <= 0.84)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.5e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.84], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.5e18

   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 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 60.1%

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

   if -4.5e18 < m < 0.839999999999999969

   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 92.3%

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

   7. Simplified 92.3%

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

   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 89.2%

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

   10. Simplified 89.2%

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

   if 0.839999999999999969 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.5e-48)
   (/ a (* k k))
   (if (<= m 1.75) (/ a (+ 1.0 (* k 10.0))) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.5e-48) {
		tmp = a / (k * k);
	} else if (m <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.5d-48)) then
        tmp = a / (k * k)
    else if (m <= 1.75d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.5e-48) {
		tmp = a / (k * k);
	} else if (m <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.5e-48:
		tmp = a / (k * k)
	elif m <= 1.75:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.5e-48)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.75)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.5e-48)
		tmp = a / (k * k);
	elseif (m <= 1.75)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.5e-48], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.75], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.49999999999999991e-48

   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 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 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 59.7%

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

   if -3.49999999999999991e-48 < m < 1.75

   1. Initial program 95.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in m around 0

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

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

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

      2. +-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 94.7%

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

   7. Simplified 94.7%

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

   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 69.0%

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

   10. Simplified 69.0%

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

   if 1.75 < m

   1. Initial program 74.7%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.7%

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

   3. Simplified 74.7%

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

   5. 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 3.0%

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

   7. Simplified 3.0%

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

   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. distribute-rgt1-in N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot \color{blue}{99} \]

       2. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot 99\right)} \]

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.24:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.12e-307) (/ a (* k k)) (if (<= k 0.24) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.12e-307) {
		tmp = a / (k * k);
	} else if (k <= 0.24) {
		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 <= 1.12d-307) then
        tmp = a / (k * k)
    else if (k <= 0.24d0) 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 <= 1.12e-307) {
		tmp = a / (k * k);
	} else if (k <= 0.24) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.12e-307:
		tmp = a / (k * k)
	elif k <= 0.24:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.12e-307)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.24)
		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 <= 1.12e-307)
		tmp = a / (k * k);
	elseif (k <= 0.24)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.12e-307], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.24], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.11999999999999994e-307

   1. Initial program 91.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in k around 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 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 33.1%

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

   if 1.11999999999999994e-307 < k < 0.23999999999999999

   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 55.0%

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

   7. Simplified 55.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 54.0%

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

   if 0.23999999999999999 < k

   1. Initial program 81.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]
   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 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. pow-lowering-pow.f64 97.4%

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

   10. Simplified 97.4%

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

   11. Taylor expanded in m around 0

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

       1. /-lowering-/.f64 65.7%

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

   13. Simplified 65.7%

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

Alternative 12: 46.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 2.2 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.24:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))) (if (<= k 2.2e-308) t_0 (if (<= k 0.24) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2.2e-308) {
		tmp = t_0;
	} else if (k <= 0.24) {
		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 <= 2.2d-308) then
        tmp = t_0
    else if (k <= 0.24d0) 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 <= 2.2e-308) {
		tmp = t_0;
	} else if (k <= 0.24) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 2.2e-308:
		tmp = t_0
	elif k <= 0.24:
		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 <= 2.2e-308)
		tmp = t_0;
	elseif (k <= 0.24)
		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 <= 2.2e-308)
		tmp = t_0;
	elseif (k <= 0.24)
		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, 2.2e-308], t$95$0, If[LessEqual[k, 0.24], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2000000000000002e-308 or 0.23999999999999999 < k

   1. Initial program 85.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 85.4%

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

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 80.3%

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

   7. Simplified 80.3%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 51.7%

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

   if 2.2000000000000002e-308 < k < 0.23999999999999999

   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 55.0%

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

   7. Simplified 55.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 54.0%

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

Alternative 13: 19.4% 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 90.7%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 90.7%

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

3. Simplified 90.7%

   \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
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 50.9%

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

7. Simplified 50.9%

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

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 22.4%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))