Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 98.4%

Time: 12.7s

Alternatives: 19

Speedup: 1.0×

• 22.3% 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: 90.2% 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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m} + k \cdot \left(\frac{k}{a\_m} + \frac{10}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ (* k 10.0) 1.0) (* k k))) 2e+102)
      (/ (pow k m) (+ (/ 1.0 a_m) (* k (+ (/ k a_m) (/ 10.0 a_m)))))
      t_0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102) {
		tmp = pow(k, m) / ((1.0 / a_m) + (k * ((k / a_m) + (10.0 / a_m))));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if ((t_0 / (((k * 10.0d0) + 1.0d0) + (k * k))) <= 2d+102) then
        tmp = (k ** m) / ((1.0d0 / a_m) + (k * ((k / a_m) + (10.0d0 / a_m))))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102) {
		tmp = Math.pow(k, m) / ((1.0 / a_m) + (k * ((k / a_m) + (10.0 / a_m))));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if (t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102:
		tmp = math.pow(k, m) / ((1.0 / a_m) + (k * ((k / a_m) + (10.0 / a_m))))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k))) <= 2e+102)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 / a_m) + Float64(k * Float64(Float64(k / a_m) + Float64(10.0 / a_m)))));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102)
		tmp = (k ^ m) / ((1.0 / a_m) + (k * ((k / a_m) + (10.0 / a_m))));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t$95$0 / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+102], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(k * N[(N[(k / a$95$m), $MachinePrecision] + N[(10.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.99999999999999995e102

   1. Initial program 97.3%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 97.3%

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

   3. Simplified 97.3%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. associate-+l+ N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 95.4%

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

   8. Applied egg-rr 95.4%

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

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

       5. +-commutative N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. /-lowering-/.f64 97.7%

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

   11. Simplified 97.7%

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

   if 1.99999999999999995e102 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 66.1%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a} + k \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m} + k \cdot \frac{k}{a\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ (* k 10.0) 1.0) (* k k))) 2e+102)
      (/ (pow k m) (+ (/ 1.0 a_m) (* k (/ k a_m))))
      t_0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102) {
		tmp = pow(k, m) / ((1.0 / a_m) + (k * (k / a_m)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if ((t_0 / (((k * 10.0d0) + 1.0d0) + (k * k))) <= 2d+102) then
        tmp = (k ** m) / ((1.0d0 / a_m) + (k * (k / a_m)))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102) {
		tmp = Math.pow(k, m) / ((1.0 / a_m) + (k * (k / a_m)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if (t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102:
		tmp = math.pow(k, m) / ((1.0 / a_m) + (k * (k / a_m)))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k))) <= 2e+102)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 / a_m) + Float64(k * Float64(k / a_m))));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 2e+102)
		tmp = (k ^ m) / ((1.0 / a_m) + (k * (k / a_m)));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t$95$0 / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+102], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(k * N[(k / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.99999999999999995e102

   1. Initial program 97.3%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 97.3%

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

   3. Simplified 97.3%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. associate-+l+ N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 95.4%

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

   8. Applied egg-rr 95.4%

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

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

       5. +-commutative N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. /-lowering-/.f64 97.7%

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

   11. Simplified 97.7%

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

   12. Taylor expanded in k around inf

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

       1. /-lowering-/.f64 96.9%

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

   14. Simplified 96.9%

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

   if 1.99999999999999995e102 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 66.1%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a} + k \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.38 \cdot 10^{-18}:\\ \;\;\;\;\frac{a\_m \cdot {k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 1.38e-18)
    (/ (* a_m (pow k m)) (+ (* k (+ k 10.0)) 1.0))
    (/ (pow k m) (/ 1.0 a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.38e-18) {
		tmp = (a_m * pow(k, m)) / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = pow(k, m) / (1.0 / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.38d-18) then
        tmp = (a_m * (k ** m)) / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = (k ** m) / (1.0d0 / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.38e-18) {
		tmp = (a_m * Math.pow(k, m)) / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = Math.pow(k, m) / (1.0 / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.38e-18:
		tmp = (a_m * math.pow(k, m)) / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = math.pow(k, m) / (1.0 / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.38e-18)
		tmp = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = Float64((k ^ m) / Float64(1.0 / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.38e-18)
		tmp = (a_m * (k ^ m)) / ((k * (k + 10.0)) + 1.0);
	else
		tmp = (k ^ m) / (1.0 / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.38e-18], N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] / N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.38e-18

   1. Initial program 96.9%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 96.9%

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

   3. Simplified 96.9%

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

   if 1.38e-18 < m

   1. Initial program 75.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 75.8%

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

   3. Simplified 75.8%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. associate-+l+ N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 67.8%

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{1}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.38 \cdot 10^{-18}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.38 \cdot 10^{-18}:\\ \;\;\;\;a\_m \cdot \frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 1.38e-18)
    (* a_m (/ (pow k m) (+ (* k (+ k 10.0)) 1.0)))
    (/ (pow k m) (/ 1.0 a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.38e-18) {
		tmp = a_m * (pow(k, m) / ((k * (k + 10.0)) + 1.0));
	} else {
		tmp = pow(k, m) / (1.0 / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.38d-18) then
        tmp = a_m * ((k ** m) / ((k * (k + 10.0d0)) + 1.0d0))
    else
        tmp = (k ** m) / (1.0d0 / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.38e-18) {
		tmp = a_m * (Math.pow(k, m) / ((k * (k + 10.0)) + 1.0));
	} else {
		tmp = Math.pow(k, m) / (1.0 / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.38e-18:
		tmp = a_m * (math.pow(k, m) / ((k * (k + 10.0)) + 1.0))
	else:
		tmp = math.pow(k, m) / (1.0 / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.38e-18)
		tmp = Float64(a_m * Float64((k ^ m) / Float64(Float64(k * Float64(k + 10.0)) + 1.0)));
	else
		tmp = Float64((k ^ m) / Float64(1.0 / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.38e-18)
		tmp = a_m * ((k ^ m) / ((k * (k + 10.0)) + 1.0));
	else
		tmp = (k ^ m) / (1.0 / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.38e-18], N[(a$95$m * N[(N[Power[k, m], $MachinePrecision] / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] / N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.38e-18

   1. Initial program 96.9%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 96.9%

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

   3. Simplified 96.9%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. associate-+l+ N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 1.38e-18 < m

   1. Initial program 75.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 75.8%

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

   3. Simplified 75.8%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. associate-+l+ N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. associate-/r/ N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 67.8%

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{1}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.38 \cdot 10^{-18}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.245:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a\_m}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= m -0.245)
      t_0
      (if (<= m 2.8e-19) (/ a_m (+ (+ (* k 10.0) 1.0) (* k k))) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if (m <= -0.245) {
		tmp = t_0;
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if (m <= (-0.245d0)) then
        tmp = t_0
    else if (m <= 2.8d-19) then
        tmp = a_m / (((k * 10.0d0) + 1.0d0) + (k * k))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if (m <= -0.245) {
		tmp = t_0;
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if m <= -0.245:
		tmp = t_0
	elif m <= 2.8e-19:
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (m <= -0.245)
		tmp = t_0;
	elseif (m <= 2.8e-19)
		tmp = Float64(a_m / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if (m <= -0.245)
		tmp = t_0;
	elseif (m <= 2.8e-19)
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -0.245], t$95$0, If[LessEqual[m, 2.8e-19], N[(a$95$m / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.245 or 2.80000000000000003e-19 < m

   1. Initial program 87.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.7%

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

   3. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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 -0.245 < m < 2.80000000000000003e-19

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 93.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.245:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.66:\\ \;\;\;\;\frac{a\_m - \frac{a\_m - \frac{a\_m}{k \cdot k}}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a\_m}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + \left(k \cdot k\right) \cdot \left(a\_m \cdot \left(k \cdot k + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.66)
    (/ (- a_m (/ (- a_m (/ a_m (* k k))) (* k k))) (* k k))
    (if (<= m 2.8e-19)
      (/ a_m (+ (+ (* k 10.0) 1.0) (* k k)))
      (+ a_m (* (* k k) (* a_m (+ (* k k) -1.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.66) {
		tmp = (a_m - ((a_m - (a_m / (k * k))) / (k * k))) / (k * k);
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.66d0)) then
        tmp = (a_m - ((a_m - (a_m / (k * k))) / (k * k))) / (k * k)
    else if (m <= 2.8d-19) then
        tmp = a_m / (((k * 10.0d0) + 1.0d0) + (k * k))
    else
        tmp = a_m + ((k * k) * (a_m * ((k * k) + (-1.0d0))))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.66) {
		tmp = (a_m - ((a_m - (a_m / (k * k))) / (k * k))) / (k * k);
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.66:
		tmp = (a_m - ((a_m - (a_m / (k * k))) / (k * k))) / (k * k)
	elif m <= 2.8e-19:
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k))
	else:
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.66)
		tmp = Float64(Float64(a_m - Float64(Float64(a_m - Float64(a_m / Float64(k * k))) / Float64(k * k))) / Float64(k * k));
	elseif (m <= 2.8e-19)
		tmp = Float64(a_m / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)));
	else
		tmp = Float64(a_m + Float64(Float64(k * k) * Float64(a_m * Float64(Float64(k * k) + -1.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.66)
		tmp = (a_m - ((a_m - (a_m / (k * k))) / (k * k))) / (k * k);
	elseif (m <= 2.8e-19)
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	else
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.66], N[(N[(a$95$m - N[(N[(a$95$m - N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.8e-19], N[(a$95$m / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(N[(k * k), $MachinePrecision] * N[(a$95$m * N[(N[(k * k), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.660000000000000031

   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. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 40.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 40.9%

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

   10. Simplified 40.9%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

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

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

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

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

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

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

       5. *-lowering-*.f64 44.2%

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

   12. Applied egg-rr 44.2%

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

   13. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + \frac{a}{{k}^{4}}\right) - \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   14. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \frac{\left(a + \frac{a}{{k}^{4}}\right) + \left(\mathsf{neg}\left(\frac{a}{{k}^{2}}\right)\right)}{{\color{blue}{k}}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\left(\frac{a}{{k}^{4}} + a\right) + \left(\mathsf{neg}\left(\frac{a}{{k}^{2}}\right)\right)}{{k}^{2}} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{a}{{k}^{4}} + a\right) + -1 \cdot \frac{a}{{k}^{2}}}{{k}^{2}} \]

       4. associate-+l+ N/A

          \[\leadsto \frac{\frac{a}{{k}^{4}} + \left(a + -1 \cdot \frac{a}{{k}^{2}}\right)}{{\color{blue}{k}}^{2}} \]

       5. remove-double-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{a}{{k}^{4}}\right)\right)\right)\right) + \left(a + -1 \cdot \frac{a}{{k}^{2}}\right)}{{k}^{2}} \]

       6. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(-1 \cdot \frac{a}{{k}^{4}}\right)\right) + \left(a + -1 \cdot \frac{a}{{k}^{2}}\right)}{{k}^{2}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-1 \cdot \frac{a}{{k}^{4}}\right)\right) + \left(a + -1 \cdot \frac{a}{{k}^{2}}\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   15. Simplified 72.4%

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

   if -0.660000000000000031 < m < 2.80000000000000003e-19

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 93.7%

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

   if 2.80000000000000003e-19 < m

   1. Initial program 76.1%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in m around 0

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

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

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

      2. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 5.1%

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

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   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 5.1%

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

   10. Simplified 5.1%

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

   11. Taylor expanded in k around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 39.9%

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

   13. Simplified 39.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.66:\\ \;\;\;\;\frac{a - \frac{a - \frac{a}{k \cdot k}}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot k\right) \cdot \left(a \cdot \left(k \cdot k + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.39:\\ \;\;\;\;\frac{a\_m - \frac{\frac{a\_m \cdot -99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a\_m}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + \left(k \cdot k\right) \cdot \left(a\_m \cdot \left(k \cdot k + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.39)
    (/ (- a_m (/ (/ (* a_m -99.0) k) k)) (* k k))
    (if (<= m 2.8e-19)
      (/ a_m (+ (+ (* k 10.0) 1.0) (* k k)))
      (+ a_m (* (* k k) (* a_m (+ (* k k) -1.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.39) {
		tmp = (a_m - (((a_m * -99.0) / k) / k)) / (k * k);
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.39d0)) then
        tmp = (a_m - (((a_m * (-99.0d0)) / k) / k)) / (k * k)
    else if (m <= 2.8d-19) then
        tmp = a_m / (((k * 10.0d0) + 1.0d0) + (k * k))
    else
        tmp = a_m + ((k * k) * (a_m * ((k * k) + (-1.0d0))))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.39) {
		tmp = (a_m - (((a_m * -99.0) / k) / k)) / (k * k);
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.39:
		tmp = (a_m - (((a_m * -99.0) / k) / k)) / (k * k)
	elif m <= 2.8e-19:
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k))
	else:
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.39)
		tmp = Float64(Float64(a_m - Float64(Float64(Float64(a_m * -99.0) / k) / k)) / Float64(k * k));
	elseif (m <= 2.8e-19)
		tmp = Float64(a_m / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)));
	else
		tmp = Float64(a_m + Float64(Float64(k * k) * Float64(a_m * Float64(Float64(k * k) + -1.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.39)
		tmp = (a_m - (((a_m * -99.0) / k) / k)) / (k * k);
	elseif (m <= 2.8e-19)
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	else
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.39], N[(N[(a$95$m - N[(N[(N[(a$95$m * -99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.8e-19], N[(a$95$m / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(N[(k * k), $MachinePrecision] * N[(a$95$m * N[(N[(k * k), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.39000000000000001

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

      2. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 40.9%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 49.3%

       \[\leadsto \color{blue}{\frac{a - \frac{a \cdot 10 - \frac{\frac{a \cdot 10}{k} + \left(\frac{-990 \cdot a}{k} + a \cdot 99\right)}{k}}{k}}{k \cdot k}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\color{blue}{\left(\frac{-99 \cdot \left(a \cdot k\right) - \left(-990 \cdot a + 10 \cdot a\right)}{{k}^{2}}\right)}, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-99 \cdot \left(a \cdot k\right) - \left(-990 \cdot a + 10 \cdot a\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-99 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(\left(-990 \cdot a + 10 \cdot a\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-99 \cdot \left(a \cdot k\right)\right), \left(\mathsf{neg}\left(\left(-990 \cdot a + 10 \cdot a\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot k\right) \cdot -99\right), \left(\mathsf{neg}\left(\left(-990 \cdot a + 10 \cdot a\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(k \cdot -99\right)\right), \left(\mathsf{neg}\left(\left(-990 \cdot a + 10 \cdot a\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(k \cdot -99\right)\right), \left(\mathsf{neg}\left(\left(-990 \cdot a + 10 \cdot a\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \left(\mathsf{neg}\left(\left(-990 \cdot a + 10 \cdot a\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \left(\mathsf{neg}\left(a \cdot \left(-990 + 10\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \left(a \cdot \left(\mathsf{neg}\left(\left(-990 + 10\right)\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \left(a \cdot \left(\mathsf{neg}\left(-980\right)\right)\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \left(a \cdot 980\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \mathsf{*.f64}\left(a, 980\right)\right), \left({k}^{2}\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \mathsf{*.f64}\left(a, 980\right)\right), \left(k \cdot k\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       14. *-lowering-*.f64 37.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, -99\right)\right), \mathsf{*.f64}\left(a, 980\right)\right), \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   13. Simplified 37.2%

       \[\leadsto \frac{a - \frac{\color{blue}{\frac{a \cdot \left(k \cdot -99\right) + a \cdot 980}{k \cdot k}}}{k}}{k \cdot k} \]

   14. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 68.9%

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

   16. Simplified 68.9%

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

   if -0.39000000000000001 < m < 2.80000000000000003e-19

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 93.7%

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

   if 2.80000000000000003e-19 < m

   1. Initial program 76.1%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in m around 0

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

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

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

      2. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 5.1%

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

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   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 5.1%

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

   10. Simplified 5.1%

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

   11. Taylor expanded in k around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 39.9%

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

   13. Simplified 39.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.39:\\ \;\;\;\;\frac{a - \frac{\frac{a \cdot -99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot k\right) \cdot \left(a \cdot \left(k \cdot k + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.29:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a\_m}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + \left(k \cdot k\right) \cdot \left(a\_m \cdot \left(k \cdot k + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.29)
    (/ 1.0 (/ (* k k) a_m))
    (if (<= m 2.3)
      (/ a_m (+ (+ (* k 10.0) 1.0) (* k k)))
      (+ a_m (* (* k k) (* a_m (+ (* k k) -1.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.29) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (m <= 2.3) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.29d0)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else if (m <= 2.3d0) then
        tmp = a_m / (((k * 10.0d0) + 1.0d0) + (k * k))
    else
        tmp = a_m + ((k * k) * (a_m * ((k * k) + (-1.0d0))))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.29) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (m <= 2.3) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.29:
		tmp = 1.0 / ((k * k) / a_m)
	elif m <= 2.3:
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k))
	else:
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.29)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	elseif (m <= 2.3)
		tmp = Float64(a_m / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)));
	else
		tmp = Float64(a_m + Float64(Float64(k * k) * Float64(a_m * Float64(Float64(k * k) + -1.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.29)
		tmp = 1.0 / ((k * k) / a_m);
	elseif (m <= 2.3)
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	else
		tmp = a_m + ((k * k) * (a_m * ((k * k) + -1.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.29], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3], N[(a$95$m / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(N[(k * k), $MachinePrecision] * N[(a$95$m * N[(N[(k * k), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.28999999999999998

   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. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 40.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 64.3%

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

   10. Simplified 64.3%

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

       1. associate-/r* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. clear-num N/A

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

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

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

       6. associate-*r/ N/A

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

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

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

       8. *-lowering-*.f64 67.6%

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

   12. Applied egg-rr 67.6%

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

   if -0.28999999999999998 < m < 2.2999999999999998

   1. Initial program 94.1%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 93.8%

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

   if 2.2999999999999998 < m

   1. Initial program 75.6%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in m around 0

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

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

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

      2. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 3.0%

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

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   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 3.0%

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

   10. Simplified 3.0%

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

   11. Taylor expanded in k around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-out N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 38.6%

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

   13. Simplified 38.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.29:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot k\right) \cdot \left(a \cdot \left(k \cdot k + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.41:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a\_m}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + k \cdot \left(k \cdot \left(a\_m \cdot 99\right) + a\_m \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.41)
    (/ 1.0 (/ (* k k) a_m))
    (if (<= m 2.8e-19)
      (/ a_m (+ (+ (* k 10.0) 1.0) (* k k)))
      (+ a_m (* k (+ (* k (* a_m 99.0)) (* a_m -10.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.41) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + (k * ((k * (a_m * 99.0)) + (a_m * -10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.41d0)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else if (m <= 2.8d-19) then
        tmp = a_m / (((k * 10.0d0) + 1.0d0) + (k * k))
    else
        tmp = a_m + (k * ((k * (a_m * 99.0d0)) + (a_m * (-10.0d0))))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.41) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (m <= 2.8e-19) {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	} else {
		tmp = a_m + (k * ((k * (a_m * 99.0)) + (a_m * -10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.41:
		tmp = 1.0 / ((k * k) / a_m)
	elif m <= 2.8e-19:
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k))
	else:
		tmp = a_m + (k * ((k * (a_m * 99.0)) + (a_m * -10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.41)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	elseif (m <= 2.8e-19)
		tmp = Float64(a_m / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)));
	else
		tmp = Float64(a_m + Float64(k * Float64(Float64(k * Float64(a_m * 99.0)) + Float64(a_m * -10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.41)
		tmp = 1.0 / ((k * k) / a_m);
	elseif (m <= 2.8e-19)
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	else
		tmp = a_m + (k * ((k * (a_m * 99.0)) + (a_m * -10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.41], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.8e-19], N[(a$95$m / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(k * N[(N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision] + N[(a$95$m * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.409999999999999976

   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. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 40.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 64.3%

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

   10. Simplified 64.3%

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

       1. associate-/r* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. clear-num N/A

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

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

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

       6. associate-*r/ N/A

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

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

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

       8. *-lowering-*.f64 67.6%

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

   12. Applied egg-rr 67.6%

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

   if -0.409999999999999976 < m < 2.80000000000000003e-19

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 93.7%

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

   if 2.80000000000000003e-19 < m

   1. Initial program 76.1%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in m around 0

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

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

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

      2. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

      22. distribute-rgt-in N/A

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

   7. Simplified 5.1%

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

   8. Taylor expanded in k around 0

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

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

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

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

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval 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 31.3%

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

   10. Simplified 31.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.41:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right) + a \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a\_m}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -3e-308)
    (/ 1.0 (/ (* k k) a_m))
    (if (<= k 10.0) (/ a_m (+ (* k 10.0) 1.0)) (/ (/ a_m k) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (k <= 10.0) {
		tmp = a_m / ((k * 10.0) + 1.0);
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-3d-308)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else if (k <= 10.0d0) then
        tmp = a_m / ((k * 10.0d0) + 1.0d0)
    else
        tmp = (a_m / k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (k <= 10.0) {
		tmp = a_m / ((k * 10.0) + 1.0);
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -3e-308:
		tmp = 1.0 / ((k * k) / a_m)
	elif k <= 10.0:
		tmp = a_m / ((k * 10.0) + 1.0)
	else:
		tmp = (a_m / k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -3e-308)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	elseif (k <= 10.0)
		tmp = Float64(a_m / Float64(Float64(k * 10.0) + 1.0));
	else
		tmp = Float64(Float64(a_m / k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -3e-308)
		tmp = 1.0 / ((k * k) / a_m);
	elseif (k <= 10.0)
		tmp = a_m / ((k * 10.0) + 1.0);
	else
		tmp = (a_m / k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -3e-308], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.0], N[(a$95$m / N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -3.00000000000000022e-308

   1. Initial program 88.6%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in 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. +-commutative N/A

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

      3. fma-define N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lft-mult-inverse N/A

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

      11. fma-define N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. lft-mult-inverse N/A

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

      15. +-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 23.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 33.9%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{1}{\frac{a}{k}}}} \]

       4. clear-num N/A

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{a}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(k \cdot \frac{k}{a}\right)}\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{k \cdot k}{\color{blue}{a}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{a}\right)\right) \]

       8. *-lowering-*.f64 34.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), a\right)\right) \]

   12. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{a}}} \]

   if -3.00000000000000022e-308 < k < 10

   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 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   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 53.4%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{10}\right)\right)\right) \]

   10. Simplified 53.4%

       \[\leadsto \frac{a}{\color{blue}{1 + k \cdot 10}} \]

   if 10 < k

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 58.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

       3. /-lowering-/.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   12. Applied egg-rr 60.7%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{elif}\;k \leq 0.14:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -3e-308)
    (/ 1.0 (/ (* k k) a_m))
    (if (<= k 0.14) (+ a_m (* a_m (* k -10.0))) (/ (/ a_m k) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (k <= 0.14) {
		tmp = a_m + (a_m * (k * -10.0));
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-3d-308)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else if (k <= 0.14d0) then
        tmp = a_m + (a_m * (k * (-10.0d0)))
    else
        tmp = (a_m / k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = 1.0 / ((k * k) / a_m);
	} else if (k <= 0.14) {
		tmp = a_m + (a_m * (k * -10.0));
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -3e-308:
		tmp = 1.0 / ((k * k) / a_m)
	elif k <= 0.14:
		tmp = a_m + (a_m * (k * -10.0))
	else:
		tmp = (a_m / k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -3e-308)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	elseif (k <= 0.14)
		tmp = Float64(a_m + Float64(a_m * Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a_m / k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -3e-308)
		tmp = 1.0 / ((k * k) / a_m);
	elseif (k <= 0.14)
		tmp = a_m + (a_m * (k * -10.0));
	else
		tmp = (a_m / k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -3e-308], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.14], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -3.00000000000000022e-308

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 88.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 23.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 33.9%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{1}{\frac{a}{k}}}} \]

       4. clear-num N/A

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{a}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(k \cdot \frac{k}{a}\right)}\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{k \cdot k}{\color{blue}{a}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{a}\right)\right) \]

       8. *-lowering-*.f64 34.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), a\right)\right) \]

   12. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{a}}} \]

   if -3.00000000000000022e-308 < k < 0.14000000000000001

   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 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{k \cdot k - 10 \cdot 10}{k - 10} \cdot k\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\color{blue}{k - 10}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(k \cdot k - 10 \cdot 10\right) \cdot k\right), \color{blue}{\left(k - 10\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k - 10 \cdot 10\right), k\right), \left(\color{blue}{k} - 10\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(100\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \mathsf{+.f64}\left(k, \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 54.1%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \mathsf{+.f64}\left(k, -10\right)\right)\right)\right) \]

   9. Applied egg-rr 54.1%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{\left(k \cdot k + -100\right) \cdot k}{k + -10}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(-10 \cdot \left(a \cdot k\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot k\right) \cdot \color{blue}{-10}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

       5. *-lowering-*.f64 53.3%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 53.3%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]

   if 0.14000000000000001 < k

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 58.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

       3. /-lowering-/.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   12. Applied egg-rr 60.7%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.14:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -3e-308)
    (/ a_m (* k k))
    (if (<= k 0.14) (+ a_m (* a_m (* k -10.0))) (/ (/ a_m k) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = a_m / (k * k);
	} else if (k <= 0.14) {
		tmp = a_m + (a_m * (k * -10.0));
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-3d-308)) then
        tmp = a_m / (k * k)
    else if (k <= 0.14d0) then
        tmp = a_m + (a_m * (k * (-10.0d0)))
    else
        tmp = (a_m / k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = a_m / (k * k);
	} else if (k <= 0.14) {
		tmp = a_m + (a_m * (k * -10.0));
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -3e-308:
		tmp = a_m / (k * k)
	elif k <= 0.14:
		tmp = a_m + (a_m * (k * -10.0))
	else:
		tmp = (a_m / k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -3e-308)
		tmp = Float64(a_m / Float64(k * k));
	elseif (k <= 0.14)
		tmp = Float64(a_m + Float64(a_m * Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a_m / k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -3e-308)
		tmp = a_m / (k * k);
	elseif (k <= 0.14)
		tmp = a_m + (a_m * (k * -10.0));
	else
		tmp = (a_m / k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -3e-308], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.14], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -3.00000000000000022e-308

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 88.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 23.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 33.9%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -3.00000000000000022e-308 < k < 0.14000000000000001

   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 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k + 10\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{k \cdot k - 10 \cdot 10}{k - 10} \cdot k\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(k \cdot k - 10 \cdot 10\right) \cdot k}{\color{blue}{k - 10}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(k \cdot k - 10 \cdot 10\right) \cdot k\right), \color{blue}{\left(k - 10\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k - 10 \cdot 10\right), k\right), \left(\color{blue}{k} - 10\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(100\right)\right)\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \left(k - 10\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \left(k + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \mathsf{+.f64}\left(k, \color{blue}{\left(\mathsf{neg}\left(10\right)\right)}\right)\right)\right)\right) \]

      13. metadata-eval 54.1%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right), k\right), \mathsf{+.f64}\left(k, -10\right)\right)\right)\right) \]

   9. Applied egg-rr 54.1%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{\left(k \cdot k + -100\right) \cdot k}{k + -10}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(-10 \cdot \left(a \cdot k\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot k\right) \cdot \color{blue}{-10}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(k \cdot -10\right)}\right)\right) \]

       5. *-lowering-*.f64 53.3%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 53.3%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]

   if 0.14000000000000001 < k

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 58.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

       3. /-lowering-/.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   12. Applied egg-rr 60.7%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\_m \cdot \left(1 - k \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -3e-308)
    (/ a_m (* k k))
    (if (<= k 1.0) (* a_m (- 1.0 (* k k))) (/ (/ a_m k) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = a_m / (k * k);
	} else if (k <= 1.0) {
		tmp = a_m * (1.0 - (k * k));
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-3d-308)) then
        tmp = a_m / (k * k)
    else if (k <= 1.0d0) then
        tmp = a_m * (1.0d0 - (k * k))
    else
        tmp = (a_m / k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = a_m / (k * k);
	} else if (k <= 1.0) {
		tmp = a_m * (1.0 - (k * k));
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -3e-308:
		tmp = a_m / (k * k)
	elif k <= 1.0:
		tmp = a_m * (1.0 - (k * k))
	else:
		tmp = (a_m / k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -3e-308)
		tmp = Float64(a_m / Float64(k * k));
	elseif (k <= 1.0)
		tmp = Float64(a_m * Float64(1.0 - Float64(k * k)));
	else
		tmp = Float64(Float64(a_m / k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -3e-308)
		tmp = a_m / (k * k);
	elseif (k <= 1.0)
		tmp = a_m * (1.0 - (k * k));
	else
		tmp = (a_m / k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -3e-308], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], N[(a$95$m * N[(1.0 - N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -3.00000000000000022e-308

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 88.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 23.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 33.9%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -3.00000000000000022e-308 < k < 1

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   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 52.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   10. Simplified 52.9%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1 + k \cdot k}{a}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + k \cdot k}{a}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + k \cdot k\right), \color{blue}{a}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(k \cdot k\right)\right), a\right)\right) \]

       5. *-lowering-*.f64 52.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), a\right)\right) \]

   12. Applied egg-rr 52.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot k}{a}}} \]

   13. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -1 \cdot \left(a \cdot {k}^{2}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto a + \left(\mathsf{neg}\left(a \cdot {k}^{2}\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto a - \color{blue}{a \cdot {k}^{2}} \]

       3. *-lft-identity N/A

          \[\leadsto 1 \cdot a - \color{blue}{a} \cdot {k}^{2} \]

       4. *-commutative N/A

          \[\leadsto 1 \cdot a - {k}^{2} \cdot \color{blue}{a} \]

       5. distribute-rgt-out-- N/A

          \[\leadsto a \cdot \color{blue}{\left(1 - {k}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 - {k}^{2}\right)}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(1, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   15. Simplified 52.9%

       \[\leadsto \color{blue}{a \cdot \left(1 - k \cdot k\right)} \]

   if 1 < k

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 58.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

       3. /-lowering-/.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   12. Applied egg-rr 60.7%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.35:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.35)
    (/ 1.0 (/ (* k k) a_m))
    (/ a_m (+ (+ (* k 10.0) 1.0) (* k k))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.35) {
		tmp = 1.0 / ((k * k) / a_m);
	} else {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.35d0)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else
        tmp = a_m / (((k * 10.0d0) + 1.0d0) + (k * k))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.35) {
		tmp = 1.0 / ((k * k) / a_m);
	} else {
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.35:
		tmp = 1.0 / ((k * k) / a_m)
	else:
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.35)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	else
		tmp = Float64(a_m / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.35)
		tmp = 1.0 / ((k * k) / a_m);
	else
		tmp = a_m / (((k * 10.0) + 1.0) + (k * k));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.35], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -0.34999999999999998

   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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 64.3%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 64.3%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{1}{\frac{a}{k}}}} \]

       4. clear-num N/A

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{a}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(k \cdot \frac{k}{a}\right)}\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{k \cdot k}{\color{blue}{a}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{a}\right)\right) \]

       8. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), a\right)\right) \]

   12. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{a}}} \]

   if -0.34999999999999998 < m

   1. Initial program 84.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 48.6%

      \[\leadsto \frac{\color{blue}{a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.35:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -3e-308) (/ a_m (* k k)) (if (<= k 1.0) a_m (/ (/ a_m k) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = a_m / (k * k);
	} else if (k <= 1.0) {
		tmp = a_m;
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-3d-308)) then
        tmp = a_m / (k * k)
    else if (k <= 1.0d0) then
        tmp = a_m
    else
        tmp = (a_m / k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -3e-308) {
		tmp = a_m / (k * k);
	} else if (k <= 1.0) {
		tmp = a_m;
	} else {
		tmp = (a_m / k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -3e-308:
		tmp = a_m / (k * k)
	elif k <= 1.0:
		tmp = a_m
	else:
		tmp = (a_m / k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -3e-308)
		tmp = Float64(a_m / Float64(k * k));
	elseif (k <= 1.0)
		tmp = a_m;
	else
		tmp = Float64(Float64(a_m / k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -3e-308)
		tmp = a_m / (k * k);
	elseif (k <= 1.0)
		tmp = a_m;
	else
		tmp = (a_m / k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -3e-308], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a$95$m, N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -3.00000000000000022e-308

   1. Initial program 88.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 88.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 23.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 33.9%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -3.00000000000000022e-308 < k < 1

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 52.9%

       \[\leadsto \color{blue}{a} \]

   if 1 < k

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 58.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

       3. /-lowering-/.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   12. Applied egg-rr 60.7%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ a_m (* k k))))
   (* a_s (if (<= k -3e-308) t_0 (if (<= k 1.0) a_m t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * k);
	double tmp;
	if (k <= -3e-308) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a_m;
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m / (k * k)
    if (k <= (-3d-308)) then
        tmp = t_0
    else if (k <= 1.0d0) then
        tmp = a_m
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * k);
	double tmp;
	if (k <= -3e-308) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a_m;
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m / (k * k)
	tmp = 0
	if k <= -3e-308:
		tmp = t_0
	elif k <= 1.0:
		tmp = a_m
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * k))
	tmp = 0.0
	if (k <= -3e-308)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a_m;
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m / (k * k);
	tmp = 0.0;
	if (k <= -3e-308)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a_m;
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -3e-308], t$95$0, If[LessEqual[k, 1.0], a$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -3.00000000000000022e-308 or 1 < k

   1. Initial program 84.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 84.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 46.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 46.5%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -3.00000000000000022e-308 < k < 1

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 52.9%

       \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.32) (/ 1.0 (/ (* k k) a_m)) (/ a_m (+ (* k (+ k 10.0)) 1.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.32) {
		tmp = 1.0 / ((k * k) / a_m);
	} else {
		tmp = a_m / ((k * (k + 10.0)) + 1.0);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.32d0)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else
        tmp = a_m / ((k * (k + 10.0d0)) + 1.0d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.32) {
		tmp = 1.0 / ((k * k) / a_m);
	} else {
		tmp = a_m / ((k * (k + 10.0)) + 1.0);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.32:
		tmp = 1.0 / ((k * k) / a_m)
	else:
		tmp = a_m / ((k * (k + 10.0)) + 1.0)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.32)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	else
		tmp = Float64(a_m / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.32)
		tmp = 1.0 / ((k * k) / a_m);
	else
		tmp = a_m / ((k * (k + 10.0)) + 1.0);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.32], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -0.320000000000000007

   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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 64.3%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 64.3%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{1}{\frac{a}{k}}}} \]

       4. clear-num N/A

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{a}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(k \cdot \frac{k}{a}\right)}\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{k \cdot k}{\color{blue}{a}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{a}\right)\right) \]

       8. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), a\right)\right) \]

   12. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{a}}} \]

   if -0.320000000000000007 < m

   1. Initial program 84.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.62:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{k \cdot k + 1}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m -0.62) (/ 1.0 (/ (* k k) a_m)) (/ a_m (+ (* k k) 1.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.62) {
		tmp = 1.0 / ((k * k) / a_m);
	} else {
		tmp = a_m / ((k * k) + 1.0);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.62d0)) then
        tmp = 1.0d0 / ((k * k) / a_m)
    else
        tmp = a_m / ((k * k) + 1.0d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.62) {
		tmp = 1.0 / ((k * k) / a_m);
	} else {
		tmp = a_m / ((k * k) + 1.0);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -0.62:
		tmp = 1.0 / ((k * k) / a_m)
	else:
		tmp = a_m / ((k * k) + 1.0)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.62)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a_m));
	else
		tmp = Float64(a_m / Float64(Float64(k * k) + 1.0));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -0.62)
		tmp = 1.0 / ((k * k) / a_m);
	else
		tmp = a_m / ((k * k) + 1.0);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.62], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -0.619999999999999996

   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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 64.3%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 64.3%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{a}{k}}{\color{blue}{k}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{1}{\frac{a}{k}}}} \]

       4. clear-num N/A

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{a}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(k \cdot \frac{k}{a}\right)}\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{k \cdot k}{\color{blue}{a}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{a}\right)\right) \]

       8. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), a\right)\right) \]

   12. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{a}}} \]

   if -0.619999999999999996 < m

   1. Initial program 84.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

      10. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

      11. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

      14. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

      22. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

   7. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   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 47.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   10. Simplified 47.7%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.62:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 19.3% accurate, 114.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot a\_m \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s a_m))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * a_m
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * a_m

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * a_m)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * a_m;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * a$95$m), $MachinePrecision]

Derivation

1. Initial program 89.8%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 89.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

3. Simplified 89.8%

   \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
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. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(10 + k\right) + \color{blue}{1}\right)\right) \]

   3. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot 1 + k, 1\right)\right)\right) \]

   5. lft-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, 10 \cdot \left(\frac{1}{k} \cdot k\right) + k, 1\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + k, 1\right)\right)\right) \]

   7. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}, 1\right)\right)\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right), 1\right)\right)\right) \]

   10. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k \cdot \left(1 + 10 \cdot \frac{1}{k}\right), \frac{1}{{k}^{2}} \cdot {k}^{2}\right)\right)\right) \]

   11. fma-define N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right) + \color{blue}{\frac{1}{{k}^{2}} \cdot {k}^{2}}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(\left(k \cdot k\right) \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \color{blue}{\frac{1}{{k}^{2}}} \cdot {k}^{2}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + \frac{\color{blue}{1}}{{k}^{2}} \cdot {k}^{2}\right)\right) \]

   14. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right) + 1\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

   18. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)}\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right)\right)\right) \]

   22. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(1 \cdot k + \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k}\right)\right)\right)\right) \]

7. Simplified 46.1%

   \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 21.2%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))