Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 96.6%

Time: 14.0s

Alternatives: 18

Speedup: 1.0×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{a\_m}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(0 - m\right)}}\\ \end{array} \end{array} \]

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 (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+63)
    (/ a_m (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
    (/ a_m (pow k (- 0.0 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 (((a_m * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+63) {
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = a_m / pow(k, (0.0 - 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 (((a_m * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 2d+63) then
        tmp = a_m / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = a_m / (k ** (0.0d0 - 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 (((a_m * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+63) {
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = a_m / Math.pow(k, (0.0 - 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 ((a_m * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+63:
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = a_m / math.pow(k, (0.0 - 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 (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+63)
		tmp = Float64(a_m / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64(a_m / (k ^ Float64(0.0 - 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 (((a_m * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+63)
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = a_m / (k ^ (0.0 - 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[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+63], N[(a$95$m / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, N[(0.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $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))) < 2.00000000000000012e63

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      9. pow-lowering-pow.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 2.00000000000000012e63 < (/.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 64.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. exp-neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

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

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

      9. /-lowering-/.f64 64.9%

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

   7. Simplified 64.9%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 100.0%

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

       1. /-rgt-identity N/A

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

       2. *-rgt-identity N/A

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

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

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

       4. inv-pow N/A

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

       5. pow-pow N/A

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

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

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

       7. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(0 - m\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(0 - m\right)}}\\ \end{array} \end{array} \]

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 -22000000000.0)
    (* a_m (pow k m))
    (if (<= m 7.5e-17)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (/ a_m (pow k (- 0.0 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 <= -22000000000.0) {
		tmp = a_m * pow(k, m);
	} else if (m <= 7.5e-17) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / pow(k, (0.0 - 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 <= (-22000000000.0d0)) then
        tmp = a_m * (k ** m)
    else if (m <= 7.5d-17) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m / (k ** (0.0d0 - 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 <= -22000000000.0) {
		tmp = a_m * Math.pow(k, m);
	} else if (m <= 7.5e-17) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / Math.pow(k, (0.0 - 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 <= -22000000000.0:
		tmp = a_m * math.pow(k, m)
	elif m <= 7.5e-17:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m / math.pow(k, (0.0 - 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 <= -22000000000.0)
		tmp = Float64(a_m * (k ^ m));
	elseif (m <= 7.5e-17)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m / (k ^ Float64(0.0 - 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 <= -22000000000.0)
		tmp = a_m * (k ^ m);
	elseif (m <= 7.5e-17)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m / (k ^ (0.0 - 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, -22000000000.0], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.5e-17], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, N[(0.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 100.0%

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

   7. Simplified 100.0%

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

   if -2.2e10 < m < 7.49999999999999984e-17

   1. Initial program 99.9%

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 99.5%

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

   7. Simplified 99.5%

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

   if 7.49999999999999984e-17 < m

   1. Initial program 77.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. exp-neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

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

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

      9. /-lowering-/.f64 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 100.0%

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

       1. /-rgt-identity N/A

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

       2. *-rgt-identity N/A

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

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

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

       4. inv-pow N/A

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

       5. pow-pow N/A

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

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

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

       7. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(0 - m\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= m -22000000000.0)
      t_0
      (if (<= m 7.5e-17) (/ a_m (+ 1.0 (* k (+ k 10.0)))) 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 <= -22000000000.0) {
		tmp = t_0;
	} else if (m <= 7.5e-17) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} 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 <= (-22000000000.0d0)) then
        tmp = t_0
    else if (m <= 7.5d-17) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    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 <= -22000000000.0) {
		tmp = t_0;
	} else if (m <= 7.5e-17) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} 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 <= -22000000000.0:
		tmp = t_0
	elif m <= 7.5e-17:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	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 <= -22000000000.0)
		tmp = t_0;
	elseif (m <= 7.5e-17)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	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 <= -22000000000.0)
		tmp = t_0;
	elseif (m <= 7.5e-17)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	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, -22000000000.0], t$95$0, If[LessEqual[m, 7.5e-17], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.2e10 or 7.49999999999999984e-17 < m

   1. Initial program 88.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.1%

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

   3. Simplified 88.1%

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

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

   if -2.2e10 < m < 7.49999999999999984e-17

   1. Initial program 99.9%

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 99.5%

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

   7. Simplified 99.5%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(0 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m \cdot {k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

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 1.0) (/ a_m (pow k (- 0.0 m))) (/ (/ (* a_m (pow k 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 <= 1.0) {
		tmp = a_m / pow(k, (0.0 - m));
	} else {
		tmp = ((a_m * pow(k, 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 <= 1.0d0) then
        tmp = a_m / (k ** (0.0d0 - m))
    else
        tmp = ((a_m * (k ** 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 <= 1.0) {
		tmp = a_m / Math.pow(k, (0.0 - m));
	} else {
		tmp = ((a_m * Math.pow(k, 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 <= 1.0:
		tmp = a_m / math.pow(k, (0.0 - m))
	else:
		tmp = ((a_m * math.pow(k, 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 <= 1.0)
		tmp = Float64(a_m / (k ^ Float64(0.0 - m)));
	else
		tmp = Float64(Float64(Float64(a_m * (k ^ 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 <= 1.0)
		tmp = a_m / (k ^ (0.0 - m));
	else
		tmp = ((a_m * (k ^ 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, 1.0], N[(a$95$m / N[Power[k, N[(0.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 94.8%

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

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

   3. Simplified 94.8%

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

   5. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. exp-neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

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

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

      9. /-lowering-/.f64 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 98.9%

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

       1. /-rgt-identity N/A

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

       2. *-rgt-identity N/A

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

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

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

       4. inv-pow N/A

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

       5. pow-pow N/A

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

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

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

       7. *-lowering-*.f64 98.9%

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

   12. Applied egg-rr 98.9%

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 98.9%

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

   14. Applied egg-rr 98.9%

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

   if 1 < k

   1. Initial program 86.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. pow-lowering-pow.f64 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{a}{{k}^{\left(0 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(0 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

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 (<= k 1.0) (/ a_m (pow k (- 0.0 m))) (* a_m (pow k (- m 2.0))))))

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 (k <= 1.0) {
		tmp = a_m / pow(k, (0.0 - m));
	} else {
		tmp = a_m * pow(k, (m - 2.0));
	}
	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 (k <= 1.0d0) then
        tmp = a_m / (k ** (0.0d0 - m))
    else
        tmp = a_m * (k ** (m - 2.0d0))
    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 (k <= 1.0) {
		tmp = a_m / Math.pow(k, (0.0 - m));
	} else {
		tmp = a_m * Math.pow(k, (m - 2.0));
	}
	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 k <= 1.0:
		tmp = a_m / math.pow(k, (0.0 - m))
	else:
		tmp = a_m * math.pow(k, (m - 2.0))
	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 (k <= 1.0)
		tmp = Float64(a_m / (k ^ Float64(0.0 - m)));
	else
		tmp = Float64(a_m * (k ^ Float64(m - 2.0)));
	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 (k <= 1.0)
		tmp = a_m / (k ^ (0.0 - m));
	else
		tmp = a_m * (k ^ (m - 2.0));
	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[k, 1.0], N[(a$95$m / N[Power[k, N[(0.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 94.8%

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

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

   3. Simplified 94.8%

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

   5. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. exp-neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

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

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

      9. /-lowering-/.f64 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 98.9%

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

       1. /-rgt-identity N/A

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

       2. *-rgt-identity N/A

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

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

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

       4. inv-pow N/A

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

       5. pow-pow N/A

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

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

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

       7. *-lowering-*.f64 98.9%

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

   12. Applied egg-rr 98.9%

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 98.9%

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

   14. Applied egg-rr 98.9%

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

   if 1 < k

   1. Initial program 86.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 85.9%

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

   7. Simplified 85.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. pow2 N/A

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

      5. pow-div N/A

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

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

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

      7. --lowering--.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{a}{{k}^{\left(0 - m\right)}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := k \cdot \left(k \cdot k\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.68:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + \frac{\frac{1000 + \frac{1000000}{t\_0}}{t\_0} - -1}{t\_0} \cdot \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}}}\\ \end{array} \end{array} \end{array} \]

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 (* k (* k k))))
   (*
    a_s
    (if (<= m -22000000000.0)
      (/ (* a_m (/ 99.0 (* k k))) (* k k))
      (if (<= m 0.68)
        (/ a_m (+ 1.0 (* k (+ k 10.0))))
        (/
         a_m
         (+
          1.0
          (*
           (/ (- (/ (+ 1000.0 (/ 1000000.0 t_0)) t_0) -1.0) t_0)
           (/
            (* k (+ (* k k) -100.0))
            (/ 1.0 (- (+ (* k k) 100.0) (* k -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 t_0 = k * (k * k);
	double tmp;
	if (m <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 0.68) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / (1.0 + (((((1000.0 + (1000000.0 / t_0)) / t_0) - -1.0) / t_0) * ((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -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) :: t_0
    real(8) :: tmp
    t_0 = k * (k * k)
    if (m <= (-22000000000.0d0)) then
        tmp = (a_m * (99.0d0 / (k * k))) / (k * k)
    else if (m <= 0.68d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m / (1.0d0 + (((((1000.0d0 + (1000000.0d0 / t_0)) / t_0) - (-1.0d0)) / t_0) * ((k * ((k * k) + (-100.0d0))) / (1.0d0 / (((k * k) + 100.0d0) - (k * (-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 t_0 = k * (k * k);
	double tmp;
	if (m <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 0.68) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / (1.0 + (((((1000.0 + (1000000.0 / t_0)) / t_0) - -1.0) / t_0) * ((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -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):
	t_0 = k * (k * k)
	tmp = 0
	if m <= -22000000000.0:
		tmp = (a_m * (99.0 / (k * k))) / (k * k)
	elif m <= 0.68:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m / (1.0 + (((((1000.0 + (1000000.0 / t_0)) / t_0) - -1.0) / t_0) * ((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -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)
	t_0 = Float64(k * Float64(k * k))
	tmp = 0.0
	if (m <= -22000000000.0)
		tmp = Float64(Float64(a_m * Float64(99.0 / Float64(k * k))) / Float64(k * k));
	elseif (m <= 0.68)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m / Float64(1.0 + Float64(Float64(Float64(Float64(Float64(1000.0 + Float64(1000000.0 / t_0)) / t_0) - -1.0) / t_0) * Float64(Float64(k * Float64(Float64(k * k) + -100.0)) / Float64(1.0 / Float64(Float64(Float64(k * k) + 100.0) - Float64(k * -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)
	t_0 = k * (k * k);
	tmp = 0.0;
	if (m <= -22000000000.0)
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	elseif (m <= 0.68)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m / (1.0 + (((((1000.0 + (1000000.0 / t_0)) / t_0) - -1.0) / t_0) * ((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -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_] := Block[{t$95$0 = N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -22000000000.0], N[(N[(a$95$m * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.68], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(1.0 + N[(N[(N[(N[(N[(1000.0 + N[(1000000.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(k * N[(N[(k * k), $MachinePrecision] + -100.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[(k * k), $MachinePrecision] + 100.0), $MachinePrecision] - N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 33.6%

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

   7. Simplified 33.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 61.3%

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

   11. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \frac{99 \cdot 1}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{99 \cdot 1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 68.7%

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

   13. Simplified 68.7%

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

   if -2.2e10 < m < 0.680000000000000049

   1. Initial program 99.8%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.9%

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

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.2%

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

   7. Simplified 98.2%

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

   if 0.680000000000000049 < m

   1. Initial program 76.5%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 2.8%

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

   7. Simplified 2.8%

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

      1. metadata-eval N/A

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

      2. sub-neg N/A

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

      3. flip-- N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. flip3-+ N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

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

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

   9. Applied egg-rr 2.3%

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

   10. Taylor expanded in k around -inf

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{1000 + 1000000 \cdot \frac{1}{{k}^{3}}}{{k}^{3}} - 1}{{k}^{3}}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), 100\right), \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{1000 + 1000000 \cdot \frac{1}{{k}^{3}}}{{k}^{3}} - 1}{{k}^{3}}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), 100\right), \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1 \cdot \frac{1000 + 1000000 \cdot \frac{1}{{k}^{3}}}{{k}^{3}} - 1}{\mathsf{neg}\left({k}^{3}\right)}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), 100\right), \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \frac{1000 + 1000000 \cdot \frac{1}{{k}^{3}}}{{k}^{3}} - 1\right), \left(\mathsf{neg}\left({k}^{3}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), -100\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, k\right), 100\right), \mathsf{*.f64}\left(k, -10\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 50.8%

       \[\leadsto \frac{a}{1 + \color{blue}{\frac{-1 - \frac{1000 + \frac{1000000}{k \cdot \left(k \cdot k\right)}}{k \cdot \left(k \cdot k\right)}}{0 - k \cdot \left(k \cdot k\right)}} \cdot \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.68:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{\frac{1000 + \frac{1000000}{k \cdot \left(k \cdot k\right)}}{k \cdot \left(k \cdot k\right)} - -1}{k \cdot \left(k \cdot k\right)} \cdot \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := k \cdot \left(k \cdot k\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.2:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}} \cdot \frac{1 + \frac{1000}{t\_0}}{t\_0}}\\ \end{array} \end{array} \end{array} \]

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 (* k (* k k))))
   (*
    a_s
    (if (<= m -22000000000.0)
      (/ (* a_m (/ 99.0 (* k k))) (* k k))
      (if (<= m 1.2)
        (/ a_m (+ 1.0 (* k (+ k 10.0))))
        (/
         a_m
         (+
          1.0
          (*
           (/
            (* k (+ (* k k) -100.0))
            (/ 1.0 (- (+ (* k k) 100.0) (* k -10.0))))
           (/ (+ 1.0 (/ 1000.0 t_0)) 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 = k * (k * k);
	double tmp;
	if (m <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 1.2) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * ((1.0 + (1000.0 / t_0)) / 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 = k * (k * k)
    if (m <= (-22000000000.0d0)) then
        tmp = (a_m * (99.0d0 / (k * k))) / (k * k)
    else if (m <= 1.2d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m / (1.0d0 + (((k * ((k * k) + (-100.0d0))) / (1.0d0 / (((k * k) + 100.0d0) - (k * (-10.0d0))))) * ((1.0d0 + (1000.0d0 / t_0)) / 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 = k * (k * k);
	double tmp;
	if (m <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 1.2) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * ((1.0 + (1000.0 / t_0)) / 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 = k * (k * k)
	tmp = 0
	if m <= -22000000000.0:
		tmp = (a_m * (99.0 / (k * k))) / (k * k)
	elif m <= 1.2:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * ((1.0 + (1000.0 / t_0)) / 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(k * Float64(k * k))
	tmp = 0.0
	if (m <= -22000000000.0)
		tmp = Float64(Float64(a_m * Float64(99.0 / Float64(k * k))) / Float64(k * k));
	elseif (m <= 1.2)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m / Float64(1.0 + Float64(Float64(Float64(k * Float64(Float64(k * k) + -100.0)) / Float64(1.0 / Float64(Float64(Float64(k * k) + 100.0) - Float64(k * -10.0)))) * Float64(Float64(1.0 + Float64(1000.0 / t_0)) / 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 = k * (k * k);
	tmp = 0.0;
	if (m <= -22000000000.0)
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	elseif (m <= 1.2)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * ((1.0 + (1000.0 / t_0)) / 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[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -22000000000.0], N[(N[(a$95$m * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.2], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(1.0 + N[(N[(N[(k * N[(N[(k * k), $MachinePrecision] + -100.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[(k * k), $MachinePrecision] + 100.0), $MachinePrecision] - N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(1000.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 33.6%

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

   7. Simplified 33.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 61.3%

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

   11. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \frac{99 \cdot 1}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{99 \cdot 1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 68.7%

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

   13. Simplified 68.7%

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

   if -2.2e10 < m < 1.19999999999999996

   1. Initial program 99.8%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.9%

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

   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. +-lowering-+.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.2%

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

   7. Simplified 98.2%

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

   if 1.19999999999999996 < m

   1. Initial program 76.5%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 2.8%

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

   7. Simplified 2.8%

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

      1. metadata-eval N/A

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

      2. sub-neg N/A

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

      3. flip-- N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. flip3-+ N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

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

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

   9. Applied egg-rr 2.3%

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

   10. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. metadata-eval N/A

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

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

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 47.7%

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

   12. Simplified 47.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}} \cdot \frac{1 + \frac{1000}{k \cdot \left(k \cdot k\right)}}{k \cdot \left(k \cdot k\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 170000000000:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}} \cdot \frac{1}{k \cdot \left(k \cdot k\right)}}\\ \end{array} \end{array} \]

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 -22000000000.0)
    (/ (* a_m (/ 99.0 (* k k))) (* k k))
    (if (<= m 170000000000.0)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (/
       a_m
       (+
        1.0
        (*
         (/ (* k (+ (* k k) -100.0)) (/ 1.0 (- (+ (* k k) 100.0) (* k -10.0))))
         (/ 1.0 (* k (* 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 <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 170000000000.0) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * (1.0 / (k * (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 <= (-22000000000.0d0)) then
        tmp = (a_m * (99.0d0 / (k * k))) / (k * k)
    else if (m <= 170000000000.0d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m / (1.0d0 + (((k * ((k * k) + (-100.0d0))) / (1.0d0 / (((k * k) + 100.0d0) - (k * (-10.0d0))))) * (1.0d0 / (k * (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 <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 170000000000.0) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * (1.0 / (k * (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 <= -22000000000.0:
		tmp = (a_m * (99.0 / (k * k))) / (k * k)
	elif m <= 170000000000.0:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * (1.0 / (k * (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 <= -22000000000.0)
		tmp = Float64(Float64(a_m * Float64(99.0 / Float64(k * k))) / Float64(k * k));
	elseif (m <= 170000000000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m / Float64(1.0 + Float64(Float64(Float64(k * Float64(Float64(k * k) + -100.0)) / Float64(1.0 / Float64(Float64(Float64(k * k) + 100.0) - Float64(k * -10.0)))) * Float64(1.0 / Float64(k * 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 <= -22000000000.0)
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	elseif (m <= 170000000000.0)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m / (1.0 + (((k * ((k * k) + -100.0)) / (1.0 / (((k * k) + 100.0) - (k * -10.0)))) * (1.0 / (k * (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, -22000000000.0], N[(N[(a$95$m * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 170000000000.0], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(1.0 + N[(N[(N[(k * N[(N[(k * k), $MachinePrecision] + -100.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[(k * k), $MachinePrecision] + 100.0), $MachinePrecision] - N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 33.6%

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

   7. Simplified 33.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 61.3%

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

   11. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \frac{99 \cdot 1}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{99 \cdot 1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 68.7%

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

   13. Simplified 68.7%

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

   if -2.2e10 < m < 1.7e11

   1. Initial program 98.8%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 96.1%

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

   7. Simplified 96.1%

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

   if 1.7e11 < m

   1. Initial program 77.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 77.1%

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

   3. Simplified 77.1%

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 2.8%

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

   7. Simplified 2.8%

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

      1. metadata-eval N/A

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

      2. sub-neg N/A

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

      3. flip-- N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. flip3-+ N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

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

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

   9. Applied egg-rr 2.3%

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

   10. Taylor expanded in k around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 33.7%

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

   12. Simplified 33.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 170000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + \frac{k \cdot \left(k \cdot k + -100\right)}{\frac{1}{\left(k \cdot k + 100\right) - k \cdot -10}} \cdot \frac{1}{k \cdot \left(k \cdot k\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m + k \cdot \left(a\_m \cdot -10 + a\_m \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -22000000000.0)
    (/ (* a_m (/ 99.0 (* k k))) (* k k))
    (if (<= m 1.18e-5)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (+ a_m (* k (+ (* a_m -10.0) (* a_m (* k 99.0)))))))))

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 <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 1.18e-5) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (k * ((a_m * -10.0) + (a_m * (k * 99.0))));
	}
	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 <= (-22000000000.0d0)) then
        tmp = (a_m * (99.0d0 / (k * k))) / (k * k)
    else if (m <= 1.18d-5) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m + (k * ((a_m * (-10.0d0)) + (a_m * (k * 99.0d0))))
    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 <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 1.18e-5) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (k * ((a_m * -10.0) + (a_m * (k * 99.0))));
	}
	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 <= -22000000000.0:
		tmp = (a_m * (99.0 / (k * k))) / (k * k)
	elif m <= 1.18e-5:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m + (k * ((a_m * -10.0) + (a_m * (k * 99.0))))
	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 <= -22000000000.0)
		tmp = Float64(Float64(a_m * Float64(99.0 / Float64(k * k))) / Float64(k * k));
	elseif (m <= 1.18e-5)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m + Float64(k * Float64(Float64(a_m * -10.0) + Float64(a_m * Float64(k * 99.0)))));
	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 <= -22000000000.0)
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	elseif (m <= 1.18e-5)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m + (k * ((a_m * -10.0) + (a_m * (k * 99.0))));
	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, -22000000000.0], N[(N[(a$95$m * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.18e-5], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(k * N[(N[(a$95$m * -10.0), $MachinePrecision] + N[(a$95$m * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 33.6%

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

   7. Simplified 33.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 61.3%

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

   11. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \frac{99 \cdot 1}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{99 \cdot 1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 68.7%

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

   13. Simplified 68.7%

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

   if -2.2e10 < m < 1.18000000000000005e-5

   1. Initial program 99.8%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.9%

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

   7. Simplified 98.9%

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

   if 1.18000000000000005e-5 < m

   1. Initial program 76.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 3.2%

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

   7. Simplified 3.2%

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

      1. +-commutative N/A

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

      2. fma-define N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. fma-define N/A

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

      6. flip-- N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. clear-num N/A

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

      12. flip-+ N/A

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

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

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

   9. Applied egg-rr 2.7%

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

   10. Taylor expanded in k around 0

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

       1. distribute-rgt-in N/A

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

       2. +-commutative N/A

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

       3. sub-neg N/A

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

       4. mul-1-neg N/A

          \[\leadsto a + \left(\left(k \cdot \left(\left(\mathsf{neg}\left(a\right)\right) + \left(\mathsf{neg}\left(-100 \cdot a\right)\right)\right)\right) \cdot k + \left(-10 \cdot a\right) \cdot k\right) \]

       5. distribute-neg-in N/A

          \[\leadsto a + \left(\left(k \cdot \left(\mathsf{neg}\left(\left(a + -100 \cdot a\right)\right)\right)\right) \cdot k + \left(-10 \cdot a\right) \cdot k\right) \]

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

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

       7. mul-1-neg N/A

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

       10. cancel-sign-sub-inv N/A

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

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

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

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

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

       13. cancel-sign-sub-inv N/A

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

   12. Simplified 26.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot -10 + a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m \cdot \frac{99}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

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 -22000000000.0)
    (/ (* a_m (/ 99.0 (* k k))) (* k k))
    (if (<= m 1.55e+44)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (+ a_m (* a_m (* k -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 <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 1.55e+44) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (a_m * (k * -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 <= (-22000000000.0d0)) then
        tmp = (a_m * (99.0d0 / (k * k))) / (k * k)
    else if (m <= 1.55d+44) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m + (a_m * (k * (-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 <= -22000000000.0) {
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	} else if (m <= 1.55e+44) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (a_m * (k * -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 <= -22000000000.0:
		tmp = (a_m * (99.0 / (k * k))) / (k * k)
	elif m <= 1.55e+44:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m + (a_m * (k * -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 <= -22000000000.0)
		tmp = Float64(Float64(a_m * Float64(99.0 / Float64(k * k))) / Float64(k * k));
	elseif (m <= 1.55e+44)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m + Float64(a_m * Float64(k * -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 <= -22000000000.0)
		tmp = (a_m * (99.0 / (k * k))) / (k * k);
	elseif (m <= 1.55e+44)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m + (a_m * (k * -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, -22000000000.0], N[(N[(a$95$m * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.55e+44], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 33.6%

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

   7. Simplified 33.6%

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

   8. Taylor expanded in k around -inf

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

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

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

   10. Simplified 61.3%

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

   11. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \frac{99 \cdot 1}{{k}^{2}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(a \cdot \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(99 \cdot \frac{1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{99 \cdot 1}{{k}^{2}}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 68.7%

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

   13. Simplified 68.7%

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

   if -2.2e10 < m < 1.54999999999999998e44

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 90.4%

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

   7. Simplified 90.4%

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

   if 1.54999999999999998e44 < m

   1. Initial program 76.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 2.8%

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

   7. Simplified 2.8%

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

      1. +-commutative N/A

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

      2. fma-define N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. fma-define N/A

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

      6. flip-- N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. clear-num N/A

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

      12. flip-+ N/A

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

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

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

   9. Applied egg-rr 2.4%

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

   10. Taylor expanded in k around 0

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

       1. associate-*r* N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 11.7%

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

   12. Simplified 11.7%

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

Alternative 11: 53.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.38 \cdot 10^{+42}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

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 -22000000000.0)
    (/ a_m (* k k))
    (if (<= m 1.38e+42)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (+ a_m (* a_m (* k -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 <= -22000000000.0) {
		tmp = a_m / (k * k);
	} else if (m <= 1.38e+42) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (a_m * (k * -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 <= (-22000000000.0d0)) then
        tmp = a_m / (k * k)
    else if (m <= 1.38d+42) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m + (a_m * (k * (-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 <= -22000000000.0) {
		tmp = a_m / (k * k);
	} else if (m <= 1.38e+42) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (a_m * (k * -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 <= -22000000000.0:
		tmp = a_m / (k * k)
	elif m <= 1.38e+42:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m + (a_m * (k * -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 <= -22000000000.0)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 1.38e+42)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m + Float64(a_m * Float64(k * -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 <= -22000000000.0)
		tmp = a_m / (k * k);
	elseif (m <= 1.38e+42)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m + (a_m * (k * -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, -22000000000.0], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.38e+42], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

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

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

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

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

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

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

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

      4. associate-+l+ N/A

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

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

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

      6. distribute-rgt-out N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 33.6%

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

   7. Simplified 33.6%

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

   8. Taylor expanded in k around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 55.2%

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

   10. Simplified 55.2%

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

   if -2.2e10 < m < 1.3800000000000001e42

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 91.3%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   if 1.3800000000000001e42 < m

   1. Initial program 76.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 76.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 2.8%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k + 10\right) + \color{blue}{1}\right)\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k + \left(\mathsf{neg}\left(-10\right)\right), 1\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k - \color{blue}{-10}, 1\right)\right)\right) \]

      5. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k - -10\right) + \color{blue}{1}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k - -10 \cdot -10}{k + -10} + 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(-10 \cdot -10\right)\right)}{k + -10} + 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(100\right)\right)}{k + -10} + 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + -100}{k + -10} + 1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{k \cdot \left(k \cdot k + -100\right)}{k + -10} + 1\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} + 1\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1}{\color{blue}{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1\right)}\right)\right) \]

   9. Applied egg-rr 2.4%

      \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right)\right) - 1}{k \cdot \left(k + 10\right) - 1}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \left(-10 \cdot a\right) \cdot \color{blue}{k} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\left(-10 \cdot a\right) \cdot k\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot -10\right) \cdot k\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

       7. *-lowering-*.f64 11.6%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 11.6%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -22000000000:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

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 -22000000000.0)
    (/ a_m (* k k))
    (if (<= m 2.2e+44) (/ a_m (+ 1.0 (* k k))) (+ a_m (* a_m (* k -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 <= -22000000000.0) {
		tmp = a_m / (k * k);
	} else if (m <= 2.2e+44) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = a_m + (a_m * (k * -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 <= (-22000000000.0d0)) then
        tmp = a_m / (k * k)
    else if (m <= 2.2d+44) then
        tmp = a_m / (1.0d0 + (k * k))
    else
        tmp = a_m + (a_m * (k * (-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 <= -22000000000.0) {
		tmp = a_m / (k * k);
	} else if (m <= 2.2e+44) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = a_m + (a_m * (k * -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 <= -22000000000.0:
		tmp = a_m / (k * k)
	elif m <= 2.2e+44:
		tmp = a_m / (1.0 + (k * k))
	else:
		tmp = a_m + (a_m * (k * -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 <= -22000000000.0)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 2.2e+44)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a_m + Float64(a_m * Float64(k * -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 <= -22000000000.0)
		tmp = a_m / (k * k);
	elseif (m <= 2.2e+44)
		tmp = a_m / (1.0 + (k * k));
	else
		tmp = a_m + (a_m * (k * -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, -22000000000.0], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.2e+44], N[(a$95$m / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.2e10

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 33.6%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      3. *-lowering-*.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 55.2%

       \[\leadsto \color{blue}{\frac{a}{k \cdot k}} \]

   if -2.2e10 < m < 2.19999999999999996e44

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   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 88.1%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   10. Simplified 88.1%

       \[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}} \]

   if 2.19999999999999996e44 < m

   1. Initial program 76.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 2.8%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k + 10\right) + \color{blue}{1}\right)\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k + \left(\mathsf{neg}\left(-10\right)\right), 1\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k - \color{blue}{-10}, 1\right)\right)\right) \]

      5. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k - -10\right) + \color{blue}{1}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k - -10 \cdot -10}{k + -10} + 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(-10 \cdot -10\right)\right)}{k + -10} + 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(100\right)\right)}{k + -10} + 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + -100}{k + -10} + 1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{k \cdot \left(k \cdot k + -100\right)}{k + -10} + 1\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} + 1\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1}{\color{blue}{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1\right)}\right)\right) \]

   9. Applied egg-rr 2.4%

      \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right)\right) - 1}{k \cdot \left(k + 10\right) - 1}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \left(-10 \cdot a\right) \cdot \color{blue}{k} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\left(-10 \cdot a\right) \cdot k\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot -10\right) \cdot k\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

       7. *-lowering-*.f64 11.7%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

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 -1.02e-84)
    (/ a_m (* k k))
    (if (<= m 1.8e+44)
      (/ a_m (+ 1.0 (* k 10.0)))
      (+ a_m (* a_m (* k -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 <= -1.02e-84) {
		tmp = a_m / (k * k);
	} else if (m <= 1.8e+44) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = a_m + (a_m * (k * -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 <= (-1.02d-84)) then
        tmp = a_m / (k * k)
    else if (m <= 1.8d+44) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = a_m + (a_m * (k * (-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 <= -1.02e-84) {
		tmp = a_m / (k * k);
	} else if (m <= 1.8e+44) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = a_m + (a_m * (k * -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 <= -1.02e-84:
		tmp = a_m / (k * k)
	elif m <= 1.8e+44:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = a_m + (a_m * (k * -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 <= -1.02e-84)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 1.8e+44)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a_m + Float64(a_m * Float64(k * -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 <= -1.02e-84)
		tmp = a_m / (k * k);
	elseif (m <= 1.8e+44)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = a_m + (a_m * (k * -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, -1.02e-84], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.8e+44], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.02000000000000004e-84

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 40.6%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      3. *-lowering-*.f64 57.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 57.7%

       \[\leadsto \color{blue}{\frac{a}{k \cdot k}} \]

   if -1.02000000000000004e-84 < m < 1.8e44

   1. Initial program 97.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 89.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + 10 \cdot k\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{10}\right)\right)\right) \]

      3. *-lowering-*.f64 70.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{10}\right)\right)\right) \]

   10. Simplified 70.7%

       \[\leadsto \frac{a}{\color{blue}{1 + k \cdot 10}} \]

   if 1.8e44 < m

   1. Initial program 76.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 2.8%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k + 10\right) + \color{blue}{1}\right)\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k + \left(\mathsf{neg}\left(-10\right)\right), 1\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k - \color{blue}{-10}, 1\right)\right)\right) \]

      5. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k - -10\right) + \color{blue}{1}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k - -10 \cdot -10}{k + -10} + 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(-10 \cdot -10\right)\right)}{k + -10} + 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(100\right)\right)}{k + -10} + 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + -100}{k + -10} + 1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{k \cdot \left(k \cdot k + -100\right)}{k + -10} + 1\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} + 1\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1}{\color{blue}{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1\right)}\right)\right) \]

   9. Applied egg-rr 2.4%

      \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right)\right) - 1}{k \cdot \left(k + 10\right) - 1}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \left(-10 \cdot a\right) \cdot \color{blue}{k} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\left(-10 \cdot a\right) \cdot k\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot -10\right) \cdot k\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

       7. *-lowering-*.f64 11.7%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

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 (<= k -8.5e-286)
    (/ a_m (* k k))
    (if (<= k 0.075) (+ a_m (* a_m (* k -10.0))) (/ a_m (* k (+ k 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 (k <= -8.5e-286) {
		tmp = a_m / (k * k);
	} else if (k <= 0.075) {
		tmp = a_m + (a_m * (k * -10.0));
	} else {
		tmp = a_m / (k * (k + 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 (k <= (-8.5d-286)) then
        tmp = a_m / (k * k)
    else if (k <= 0.075d0) then
        tmp = a_m + (a_m * (k * (-10.0d0)))
    else
        tmp = a_m / (k * (k + 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 (k <= -8.5e-286) {
		tmp = a_m / (k * k);
	} else if (k <= 0.075) {
		tmp = a_m + (a_m * (k * -10.0));
	} else {
		tmp = a_m / (k * (k + 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 k <= -8.5e-286:
		tmp = a_m / (k * k)
	elif k <= 0.075:
		tmp = a_m + (a_m * (k * -10.0))
	else:
		tmp = a_m / (k * (k + 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 (k <= -8.5e-286)
		tmp = Float64(a_m / Float64(k * k));
	elseif (k <= 0.075)
		tmp = Float64(a_m + Float64(a_m * Float64(k * -10.0)));
	else
		tmp = Float64(a_m / Float64(k * Float64(k + 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 (k <= -8.5e-286)
		tmp = a_m / (k * k);
	elseif (k <= 0.075)
		tmp = a_m + (a_m * (k * -10.0));
	else
		tmp = a_m / (k * (k + 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[k, -8.5e-286], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -8.4999999999999998e-286

   1. Initial program 88.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 22.6%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 22.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      3. *-lowering-*.f64 29.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 29.9%

       \[\leadsto \color{blue}{\frac{a}{k \cdot k}} \]

   if -8.4999999999999998e-286 < k < 0.0749999999999999972

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 51.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k + 10\right) + \color{blue}{1}\right)\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k + \left(\mathsf{neg}\left(-10\right)\right), 1\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k - \color{blue}{-10}, 1\right)\right)\right) \]

      5. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k - -10\right) + \color{blue}{1}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k - -10 \cdot -10}{k + -10} + 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(-10 \cdot -10\right)\right)}{k + -10} + 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(100\right)\right)}{k + -10} + 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + -100}{k + -10} + 1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{k \cdot \left(k \cdot k + -100\right)}{k + -10} + 1\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} + 1\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1}{\color{blue}{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1\right)}\right)\right) \]

   9. Applied egg-rr 51.0%

      \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right)\right) - 1}{k \cdot \left(k + 10\right) - 1}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \left(-10 \cdot a\right) \cdot \color{blue}{k} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\left(-10 \cdot a\right) \cdot k\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot -10\right) \cdot k\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

       7. *-lowering-*.f64 50.3%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 50.3%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]

   if 0.0749999999999999972 < k

   1. Initial program 86.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 86.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 86.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 61.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot \frac{1}{k}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left(\frac{10 \cdot 1}{\color{blue}{k}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left(\frac{10}{k}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 60.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(10, \color{blue}{k}\right)\right)\right)\right) \]

   10. Simplified 60.7%

       \[\leadsto \frac{a}{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \frac{10}{k}\right)}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right) \]

       3. +-lowering-+.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right) \]

   13. Simplified 60.7%

       \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.0076:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* k k))))
   (*
    a_s
    (if (<= k -8.5e-286)
      t_0
      (if (<= k 0.0076) (+ a_m (* a_m (* k -10.0))) 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 <= -8.5e-286) {
		tmp = t_0;
	} else if (k <= 0.0076) {
		tmp = a_m + (a_m * (k * -10.0));
	} 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 <= (-8.5d-286)) then
        tmp = t_0
    else if (k <= 0.0076d0) then
        tmp = a_m + (a_m * (k * (-10.0d0)))
    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 <= -8.5e-286) {
		tmp = t_0;
	} else if (k <= 0.0076) {
		tmp = a_m + (a_m * (k * -10.0));
	} 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 <= -8.5e-286:
		tmp = t_0
	elif k <= 0.0076:
		tmp = a_m + (a_m * (k * -10.0))
	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 <= -8.5e-286)
		tmp = t_0;
	elseif (k <= 0.0076)
		tmp = Float64(a_m + Float64(a_m * Float64(k * -10.0)));
	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 <= -8.5e-286)
		tmp = t_0;
	elseif (k <= 0.0076)
		tmp = a_m + (a_m * (k * -10.0));
	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, -8.5e-286], t$95$0, If[LessEqual[k, 0.0076], N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -8.4999999999999998e-286 or 0.00759999999999999998 < k

   1. Initial program 87.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 43.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      3. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto \color{blue}{\frac{a}{k \cdot k}} \]

   if -8.4999999999999998e-286 < k < 0.00759999999999999998

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 51.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k + 10\right) + \color{blue}{1}\right)\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k + \left(\mathsf{neg}\left(-10\right)\right), 1\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(\mathsf{fma}\left(k, k - \color{blue}{-10}, 1\right)\right)\right) \]

      5. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \left(k - -10\right) + \color{blue}{1}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k - -10 \cdot -10}{k + -10} + 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(-10 \cdot -10\right)\right)}{k + -10} + 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + \left(\mathsf{neg}\left(100\right)\right)}{k + -10} + 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \frac{k \cdot k + -100}{k + -10} + 1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{k \cdot \left(k \cdot k + -100\right)}{k + -10} + 1\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} + 1\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(\frac{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1}{\color{blue}{\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} \cdot \frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\frac{k + -10}{k \cdot \left(k \cdot k + -100\right)}} - 1\right)}\right)\right) \]

   9. Applied egg-rr 51.0%

      \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right)\right) - 1}{k \cdot \left(k + 10\right) - 1}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \left(-10 \cdot a\right) \cdot \color{blue}{k} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\left(-10 \cdot a\right) \cdot k\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(\left(a \cdot -10\right) \cdot k\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \left(a \cdot \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(-10 \cdot k\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \left(k \cdot \color{blue}{-10}\right)\right)\right) \]

       7. *-lowering-*.f64 50.3%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{-10}\right)\right)\right) \]

   12. Simplified 50.3%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -8.5e-286) 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 <= -8.5e-286) {
		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 <= (-8.5d-286)) 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 <= -8.5e-286) {
		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 <= -8.5e-286:
		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 <= -8.5e-286)
		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 <= -8.5e-286)
		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, -8.5e-286], 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 < -8.4999999999999998e-286 or 1 < k

   1. Initial program 87.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 42.6%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      3. *-lowering-*.f64 45.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 45.9%

       \[\leadsto \color{blue}{\frac{a}{k \cdot k}} \]

   if -8.4999999999999998e-286 < k < 1

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 51.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 49.5%

       \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 27.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot 10}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -2.52 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 10.0))))
   (* a_s (if (<= k -2.52e+98) t_0 (if (<= k 0.1) 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 * 10.0);
	double tmp;
	if (k <= -2.52e+98) {
		tmp = t_0;
	} else if (k <= 0.1) {
		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 * 10.0d0)
    if (k <= (-2.52d+98)) then
        tmp = t_0
    else if (k <= 0.1d0) 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 * 10.0);
	double tmp;
	if (k <= -2.52e+98) {
		tmp = t_0;
	} else if (k <= 0.1) {
		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 * 10.0)
	tmp = 0
	if k <= -2.52e+98:
		tmp = t_0
	elif k <= 0.1:
		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 * 10.0))
	tmp = 0.0
	if (k <= -2.52e+98)
		tmp = t_0;
	elseif (k <= 0.1)
		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 * 10.0);
	tmp = 0.0;
	if (k <= -2.52e+98)
		tmp = t_0;
	elseif (k <= 0.1)
		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 * 10.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -2.52e+98], t$95$0, If[LessEqual[k, 0.1], a$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -2.51999999999999999e98 or 0.10000000000000001 < k

   1. Initial program 82.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 58.3%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot \frac{1}{k}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left(\frac{10 \cdot 1}{\color{blue}{k}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left(\frac{10}{k}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 57.6%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(10, \color{blue}{k}\right)\right)\right)\right) \]

   10. Simplified 57.6%

       \[\leadsto \frac{a}{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \frac{10}{k}\right)}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(10 \cdot k\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{10}\right)\right) \]

       2. *-lowering-*.f64 28.6%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{10}\right)\right) \]

   13. Simplified 28.6%

       \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]

   if -2.51999999999999999e98 < k < 0.10000000000000001

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a \cdot {k}^{m}\right), \color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({k}^{m}\right)\right), \left(\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(\left(1 + \color{blue}{10 \cdot k}\right) + k \cdot k\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \left(1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(10 \cdot k + k \cdot k\right)}\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + k \cdot \left(10 + k\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 36.4%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

   7. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   9. Step-by-step derivation

   10. Simplified 35.7%

       \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 19.7% 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 92.1%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 92.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, m\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

3. Simplified 92.1%

   \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \]
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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(k \cdot \left(10 + k\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(10 + k\right)}\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k + \color{blue}{10}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 46.1%

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(k, \color{blue}{10}\right)\right)\right)\right) \]

7. Simplified 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 21.9%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))