Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.0%

Time: 13.3s

Alternatives: 19

Speedup: 1.2×

• Could not converge on a ground truth

  1. a = 5.503147844059505e-6
  2. k = -2.181105680002342e-48
  3. m = 5.62328258619792e+302

• 22.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{\left(m \cdot 0.5\right)}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{k} \cdot \frac{a \cdot t\_0}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (pow k (* m 0.5))))
   (if (<= k 1.0) (* a (pow k m)) (* (/ t_0 k) (/ (* a t_0) k)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, (m * 0.5));
	double tmp;
	if (k <= 1.0) {
		tmp = a * pow(k, m);
	} else {
		tmp = (t_0 / k) * ((a * t_0) / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k ** (m * 0.5d0)
    if (k <= 1.0d0) then
        tmp = a * (k ** m)
    else
        tmp = (t_0 / k) * ((a * t_0) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, (m * 0.5));
	double tmp;
	if (k <= 1.0) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (t_0 / k) * ((a * t_0) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, (m * 0.5))
	tmp = 0
	if k <= 1.0:
		tmp = a * math.pow(k, m)
	else:
		tmp = (t_0 / k) * ((a * t_0) / k)
	return tmp

Julia

function code(a, k, m)
	t_0 = k ^ Float64(m * 0.5)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64(t_0 / k) * Float64(Float64(a * t_0) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k ^ (m * 0.5);
	tmp = 0.0;
	if (k <= 1.0)
		tmp = a * (k ^ m);
	else
		tmp = (t_0 / k) * ((a * t_0) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Power[k, N[(m * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / k), $MachinePrecision] * N[(N[(a * t$95$0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 93.7%

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

   3. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 98.2

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

   5. Simplified 98.2%

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

   if 1 < k

   1. Initial program 76.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.8

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

   5. Simplified 75.8%

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

      1. *-commutative N/A

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

      2. sqr-pow N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

      8. div-inv N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. div-inv N/A

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

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

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

      16. metadata-eval 99.2

          \[\leadsto \frac{{k}^{\left(m \cdot 0.5\right)}}{k} \cdot \frac{{k}^{\left(m \cdot \color{blue}{0.5}\right)} \cdot a}{k} \]

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{{k}^{\left(m \cdot 0.5\right)}}{k} \cdot \frac{{k}^{\left(m \cdot 0.5\right)} \cdot a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{\left(m \cdot 0.5\right)}}{k} \cdot \frac{a \cdot {k}^{\left(m \cdot 0.5\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{\frac{100}{k} + \left(10 + \frac{1000}{k \cdot k}\right)}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, \mathsf{fma}\left(a, \frac{9801}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}, a\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/
      a
      (fma
       k
       (*
        (fma k k -100.0)
        (/ (+ (/ (+ (/ 100.0 k) (+ 10.0 (/ 1000.0 (* k k)))) k) 1.0) k))
       1.0))
     (if (<= t_0 1e+301)
       (/ a (+ (* k (+ k 10.0)) 1.0))
       (if (<= t_0 INFINITY)
         (/
          (fma a (/ 99.0 (* k k)) (fma a (/ 9801.0 (* k (* k (* k k)))) a))
          (* k k))
         (fma k (* a (fma 100.0 k -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(k, (fma(k, k, -100.0) * (((((100.0 / k) + (10.0 + (1000.0 / (k * k)))) / k) + 1.0) / k)), 1.0);
	} else if (t_0 <= 1e+301) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(a, (99.0 / (k * k)), fma(a, (9801.0 / (k * (k * (k * k)))), a)) / (k * k);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(k, Float64(fma(k, k, -100.0) * Float64(Float64(Float64(Float64(Float64(100.0 / k) + Float64(10.0 + Float64(1000.0 / Float64(k * k)))) / k) + 1.0) / k)), 1.0));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(a, Float64(99.0 / Float64(k * k)), fma(a, Float64(9801.0 / Float64(k * Float64(k * Float64(k * k)))), a)) / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(N[(k * k + -100.0), $MachinePrecision] * N[(N[(N[(N[(N[(100.0 / k), $MachinePrecision] + N[(10.0 + N[(1000.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(a * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(a * N[(9801.0 / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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))) < 0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 47.6

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

   5. Simplified 47.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 47.6

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

   7. Applied egg-rr 47.6%

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

   8. Taylor expanded in k around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

   10. Simplified 54.5%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 3.3

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

   7. Applied egg-rr 3.3%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 1.1

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 1.1%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + 9801 \cdot \frac{a}{{k}^{4}}\right) - -99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\left(a + 9801 \cdot \frac{a}{{k}^{4}}\right) - -99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

   13. Simplified 61.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, \mathsf{fma}\left(a, \frac{9801}{\left(k \cdot \left(k \cdot k\right)\right) \cdot k}, a\right)\right)}{k \cdot k}} \]

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. metadata-eval N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

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

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

       9. accelerator-lowering-fma.f64 86.1

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

   13. Simplified 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{\frac{100}{k} + \left(10 + \frac{1000}{k \cdot k}\right)}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, \mathsf{fma}\left(a, \frac{9801}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}, a\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10 + \frac{100}{k}}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, \mathsf{fma}\left(a, \frac{9801}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}, a\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/
      a
      (fma
       k
       (* (fma k k -100.0) (/ (+ (/ (+ 10.0 (/ 100.0 k)) k) 1.0) k))
       1.0))
     (if (<= t_0 1e+301)
       (/ a (+ (* k (+ k 10.0)) 1.0))
       (if (<= t_0 INFINITY)
         (/
          (fma a (/ 99.0 (* k k)) (fma a (/ 9801.0 (* k (* k (* k k)))) a))
          (* k k))
         (fma k (* a (fma 100.0 k -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(k, (fma(k, k, -100.0) * ((((10.0 + (100.0 / k)) / k) + 1.0) / k)), 1.0);
	} else if (t_0 <= 1e+301) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(a, (99.0 / (k * k)), fma(a, (9801.0 / (k * (k * (k * k)))), a)) / (k * k);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(k, Float64(fma(k, k, -100.0) * Float64(Float64(Float64(Float64(10.0 + Float64(100.0 / k)) / k) + 1.0) / k)), 1.0));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(a, Float64(99.0 / Float64(k * k)), fma(a, Float64(9801.0 / Float64(k * Float64(k * Float64(k * k)))), a)) / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(N[(k * k + -100.0), $MachinePrecision] * N[(N[(N[(N[(10.0 + N[(100.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(a * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(a * N[(9801.0 / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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))) < 0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 47.6

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

   5. Simplified 47.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 47.6

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

   7. Applied egg-rr 47.6%

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

   8. Taylor expanded in k around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. neg-lowering-neg.f64 53.0

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

   10. Simplified 53.0%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 3.3

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

   7. Applied egg-rr 3.3%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 1.1

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 1.1%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + 9801 \cdot \frac{a}{{k}^{4}}\right) - -99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\left(a + 9801 \cdot \frac{a}{{k}^{4}}\right) - -99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

   13. Simplified 61.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, \mathsf{fma}\left(a, \frac{9801}{\left(k \cdot \left(k \cdot k\right)\right) \cdot k}, a\right)\right)}{k \cdot k}} \]

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. metadata-eval N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

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

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

       9. accelerator-lowering-fma.f64 86.1

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

   13. Simplified 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10 + \frac{100}{k}}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, \mathsf{fma}\left(a, \frac{9801}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}, a\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10 + \frac{100}{k}}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/
      a
      (fma
       k
       (* (fma k k -100.0) (/ (+ (/ (+ 10.0 (/ 100.0 k)) k) 1.0) k))
       1.0))
     (if (<= t_0 1e+301)
       (/ a (+ (* k (+ k 10.0)) 1.0))
       (if (<= t_0 INFINITY)
         (/ (fma a (/ 99.0 (* k k)) a) (* k k))
         (fma k (* a (fma 100.0 k -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(k, (fma(k, k, -100.0) * ((((10.0 + (100.0 / k)) / k) + 1.0) / k)), 1.0);
	} else if (t_0 <= 1e+301) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(a, (99.0 / (k * k)), a) / (k * k);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(k, Float64(fma(k, k, -100.0) * Float64(Float64(Float64(Float64(10.0 + Float64(100.0 / k)) / k) + 1.0) / k)), 1.0));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(a, Float64(99.0 / Float64(k * k)), a) / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(N[(k * k + -100.0), $MachinePrecision] * N[(N[(N[(N[(10.0 + N[(100.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(a * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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))) < 0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 47.6

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

   5. Simplified 47.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 47.6

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

   7. Applied egg-rr 47.6%

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

   8. Taylor expanded in k around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. neg-lowering-neg.f64 53.0

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

   10. Simplified 53.0%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 3.3

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

   7. Applied egg-rr 3.3%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 1.1

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 1.1%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{a + 99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{99 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{2}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{99 \cdot a}{{k}^{2}}} + a}{{k}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 99}}{{k}^{2}} + a}{{k}^{2}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{99}{{k}^{2}}} + a}{{k}^{2}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{99 \cdot 1}}{{k}^{2}} + a}{{k}^{2}} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(99 \cdot \frac{1}{{k}^{2}}\right)} + a}{{k}^{2}} \]

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

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{99}}{{k}^{2}}, a\right)}{{k}^{2}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{99}{{k}^{2}}}, a\right)}{{k}^{2}} \]

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{\color{blue}{k \cdot k}} \]

       15. *-lowering-*.f64 50.1

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{\color{blue}{k \cdot k}} \]

   13. Simplified 50.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}} \]

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. metadata-eval N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

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

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

       9. accelerator-lowering-fma.f64 86.1

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

   13. Simplified 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10 + \frac{100}{k}}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (fma k (* (fma k k -100.0) (/ (+ (/ 10.0 k) 1.0) k)) 1.0))
     (if (<= t_0 1e+301)
       (/ a (+ (* k (+ k 10.0)) 1.0))
       (if (<= t_0 INFINITY)
         (/ (fma a (/ 99.0 (* k k)) a) (* k k))
         (fma k (* a (fma 100.0 k -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(k, (fma(k, k, -100.0) * (((10.0 / k) + 1.0) / k)), 1.0);
	} else if (t_0 <= 1e+301) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(a, (99.0 / (k * k)), a) / (k * k);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(k, Float64(fma(k, k, -100.0) * Float64(Float64(Float64(10.0 / k) + 1.0) / k)), 1.0));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(a, Float64(99.0 / Float64(k * k)), a) / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(N[(k * k + -100.0), $MachinePrecision] * N[(N[(N[(10.0 / k), $MachinePrecision] + 1.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(a * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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))) < 0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 47.6

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

   5. Simplified 47.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 47.6

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

   7. Applied egg-rr 47.6%

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

   8. Taylor expanded in k around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 49.7

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

   10. Simplified 49.7%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 3.3

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

   7. Applied egg-rr 3.3%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 1.1

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 1.1%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{a + 99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{99 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{2}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{99 \cdot a}{{k}^{2}}} + a}{{k}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 99}}{{k}^{2}} + a}{{k}^{2}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{99}{{k}^{2}}} + a}{{k}^{2}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{99 \cdot 1}}{{k}^{2}} + a}{{k}^{2}} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(99 \cdot \frac{1}{{k}^{2}}\right)} + a}{{k}^{2}} \]

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

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{99}}{{k}^{2}}, a\right)}{{k}^{2}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{99}{{k}^{2}}}, a\right)}{{k}^{2}} \]

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{\color{blue}{k \cdot k}} \]

       15. *-lowering-*.f64 50.1

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{\color{blue}{k \cdot k}} \]

   13. Simplified 50.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}} \]

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. sub-neg N/A

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

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

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

      12. metadata-eval 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. metadata-eval N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. distribute-rgt-out N/A

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

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

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

       9. accelerator-lowering-fma.f64 86.1

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

   13. Simplified 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{k \cdot k}, k\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* k (fma k (/ -99.0 (* k k)) k)))
     (if (<= t_0 1e+301)
       (/ a (+ (* k (+ k 10.0)) 1.0))
       (if (<= t_0 INFINITY)
         (/ (fma a (/ 99.0 (* k k)) a) (* k k))
         (fma k (* a (fma 100.0 k -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (k * fma(k, (-99.0 / (k * k)), k));
	} else if (t_0 <= 1e+301) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(a, (99.0 / (k * k)), a) / (k * k);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(k * fma(k, Float64(-99.0 / Float64(k * k)), k)));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(a, Float64(99.0 / Float64(k * k)), a) / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(k * N[(-99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(a * N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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))) < 0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 47.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 47.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 47.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 34.6

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 34.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{a}{\color{blue}{{k}^{2} \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{a}{\color{blue}{\left(k \cdot k\right)} \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)} \]

       2. associate-*l* N/A

          \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)\right)}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)\right)}} \]

       4. sub-neg N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right)\right)}\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(\left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right) + 1\right)}\right)} \]

       6. distribute-lft-in N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right) + k \cdot 1\right)}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right) + \color{blue}{k}\right)} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\mathsf{fma}\left(k, \mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right), k\right)}} \]

       9. associate-*r/ N/A

          \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{\frac{99 \cdot 1}{{k}^{2}}}\right), k\right)} \]

       10. metadata-eval N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \mathsf{neg}\left(\frac{\color{blue}{99}}{{k}^{2}}\right), k\right)} \]

       11. distribute-neg-frac N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \color{blue}{\frac{\mathsf{neg}\left(99\right)}{{k}^{2}}}, k\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \frac{\color{blue}{-99}}{{k}^{2}}, k\right)} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \color{blue}{\frac{-99}{{k}^{2}}}, k\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{\color{blue}{k \cdot k}}, k\right)} \]

       15. *-lowering-*.f64 48.0

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{\color{blue}{k \cdot k}}, k\right)} \]

   13. Simplified 48.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot \mathsf{fma}\left(k, \frac{-99}{k \cdot k}, k\right)}} \]

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 99.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

      4. +-lowering-+.f64 99.9

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right) + 1}} \]

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 3.3

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.3%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 3.3

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 3.3%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 1.1

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 1.1%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + 99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{99 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{2}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{99 \cdot a}{{k}^{2}}} + a}{{k}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 99}}{{k}^{2}} + a}{{k}^{2}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{99}{{k}^{2}}} + a}{{k}^{2}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{99 \cdot 1}}{{k}^{2}} + a}{{k}^{2}} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(99 \cdot \frac{1}{{k}^{2}}\right)} + a}{{k}^{2}} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, 99 \cdot \frac{1}{{k}^{2}}, a\right)}}{{k}^{2}} \]

       9. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{99 \cdot 1}{{k}^{2}}}, a\right)}{{k}^{2}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{99}}{{k}^{2}}, a\right)}{{k}^{2}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{99}{{k}^{2}}}, a\right)}{{k}^{2}} \]

       12. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{\color{blue}{k \cdot k}}, a\right)}{{k}^{2}} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{\color{blue}{k \cdot k}}, a\right)}{{k}^{2}} \]

       14. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{\color{blue}{k \cdot k}} \]

       15. *-lowering-*.f64 50.1

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{\color{blue}{k \cdot k}} \]

   13. Simplified 50.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}} \]

   if +inf.0 < (/.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 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 1.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 1.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 1.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1}{10}}, 1\right)} \]
   9. Step-by-step derivation

   10. Simplified 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right) + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) - 10 \cdot a, a\right)} \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{100 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a, a\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(100 \cdot k\right) \cdot a} + -10 \cdot a, a\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       9. accelerator-lowering-fma.f64 86.1

          \[\leadsto \mathsf{fma}\left(k, a \cdot \color{blue}{\mathsf{fma}\left(100, k, -10\right)}, a\right) \]

   13. Simplified 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{k \cdot k}, k\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{99}{k \cdot k}, a\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{k \cdot k}, k\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* k (fma k (/ -99.0 (* k k)) k)))
     (if (<= t_0 1e+301)
       (/ a (+ (* k (+ k 10.0)) 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma k (* a (fma 100.0 k -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (k * fma(k, (-99.0 / (k * k)), k));
	} else if (t_0 <= 1e+301) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(k * fma(k, Float64(-99.0 / Float64(k * k)), k)));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(k * N[(-99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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))) < 0.0

   1. Initial program 96.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 47.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 47.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 47.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 34.6

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   10. Simplified 34.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{a}{\color{blue}{{k}^{2} \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)}} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{a}{\color{blue}{\left(k \cdot k\right)} \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)} \]

       2. associate-*l* N/A

          \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)\right)}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k \cdot \left(1 - 99 \cdot \frac{1}{{k}^{2}}\right)\right)}} \]

       4. sub-neg N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right)\right)}\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(\left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right) + 1\right)}\right)} \]

       6. distribute-lft-in N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right) + k \cdot 1\right)}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \left(\mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right)\right) + \color{blue}{k}\right)} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\mathsf{fma}\left(k, \mathsf{neg}\left(99 \cdot \frac{1}{{k}^{2}}\right), k\right)}} \]

       9. associate-*r/ N/A

          \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{\frac{99 \cdot 1}{{k}^{2}}}\right), k\right)} \]

       10. metadata-eval N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \mathsf{neg}\left(\frac{\color{blue}{99}}{{k}^{2}}\right), k\right)} \]

       11. distribute-neg-frac N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \color{blue}{\frac{\mathsf{neg}\left(99\right)}{{k}^{2}}}, k\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \frac{\color{blue}{-99}}{{k}^{2}}, k\right)} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \color{blue}{\frac{-99}{{k}^{2}}}, k\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{\color{blue}{k \cdot k}}, k\right)} \]

       15. *-lowering-*.f64 48.0

           \[\leadsto \frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{\color{blue}{k \cdot k}}, k\right)} \]

   13. Simplified 48.0%

       \[\leadsto \frac{a}{\color{blue}{k \cdot \mathsf{fma}\left(k, \frac{-99}{k \cdot k}, k\right)}} \]

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 99.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

      4. +-lowering-+.f64 99.9

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right) + 1}} \]

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 3.3

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.3%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 45.4

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 45.4%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if +inf.0 < (/.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 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 1.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 1.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 1.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1}{10}}, 1\right)} \]
   9. Step-by-step derivation

   10. Simplified 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right) + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) - 10 \cdot a, a\right)} \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{100 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a, a\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(100 \cdot k\right) \cdot a} + -10 \cdot a, a\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       9. accelerator-lowering-fma.f64 86.1

          \[\leadsto \mathsf{fma}\left(k, a \cdot \color{blue}{\mathsf{fma}\left(100, k, -10\right)}, a\right) \]

   13. Simplified 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{k \cdot \mathsf{fma}\left(k, \frac{-99}{k \cdot k}, k\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+244}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ (* k 10.0) 1.0) (* k k))) 1e+244)
     (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / (((k * 10.0) + 1.0) + (k * k))) <= 1e+244) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k))) <= 1e+244)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+244], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000007e244

   1. Initial program 97.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot a \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a \]

      6. associate-+l+ N/A

         \[\leadsto \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \cdot a \]

      8. distribute-rgt-out N/A

         \[\leadsto \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \cdot a \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \cdot a \]

      10. +-lowering-+.f64 97.0

          \[\leadsto \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \cdot a \]

   4. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\mathsf{fma}\left(k, 10 + k, 1\right)} \cdot a} \]

   if 1.00000000000000007e244 < (/.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 48.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. pow-lowering-pow.f64 100.0

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 10^{+244}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.0011:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -0.0011)
     t_0
     (if (<= m 1.6e-25) (/ a (+ (* k (+ k 10.0)) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -0.0011) {
		tmp = t_0;
	} else if (m <= 1.6e-25) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-0.0011d0)) then
        tmp = t_0
    else if (m <= 1.6d-25) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -0.0011) {
		tmp = t_0;
	} else if (m <= 1.6e-25) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -0.0011:
		tmp = t_0
	elif m <= 1.6e-25:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -0.0011)
		tmp = t_0;
	elseif (m <= 1.6e-25)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -0.0011)
		tmp = t_0;
	elseif (m <= 1.6e-25)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.0011], t$95$0, If[LessEqual[m, 1.6e-25], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -0.00110000000000000007 or 1.6000000000000001e-25 < m

   1. Initial program 83.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. pow-lowering-pow.f64 99.4

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   if -0.00110000000000000007 < m < 1.6000000000000001e-25

   1. Initial program 93.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 93.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

      4. +-lowering-+.f64 93.1

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

   7. Applied egg-rr 93.1%

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right) + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 97.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.0) (* a (pow k m)) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.0d0) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m + -2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m + -2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m + -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 93.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. pow-lowering-pow.f64 98.2

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   if 1 < k

   1. Initial program 76.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 75.8

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]

   5. Simplified 75.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{k \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{k \cdot k} \cdot a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{k \cdot k} \cdot a} \]

      4. pow2 N/A

         \[\leadsto \frac{{k}^{m}}{\color{blue}{{k}^{2}}} \cdot a \]

      5. pow-div N/A

         \[\leadsto \color{blue}{{k}^{\left(m - 2\right)}} \cdot a \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{k}^{\left(m - 2\right)}} \cdot a \]

      7. sub-neg N/A

         \[\leadsto {k}^{\color{blue}{\left(m + \left(\mathsf{neg}\left(2\right)\right)\right)}} \cdot a \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {k}^{\color{blue}{\left(m + \left(\mathsf{neg}\left(2\right)\right)\right)}} \cdot a \]

      9. metadata-eval 92.9

         \[\leadsto {k}^{\left(m + \color{blue}{-2}\right)} \cdot a \]

   7. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{{k}^{\left(m + -2\right)} \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -2.05 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -1.85 \cdot 10^{-264}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1800000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -2.05e-107)
     t_0
     (if (<= m -1.85e-264)
       a
       (if (<= m 1800000000000.0) t_0 (* -10.0 (* k a)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -2.05e-107) {
		tmp = t_0;
	} else if (m <= -1.85e-264) {
		tmp = a;
	} else if (m <= 1800000000000.0) {
		tmp = t_0;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-2.05d-107)) then
        tmp = t_0
    else if (m <= (-1.85d-264)) then
        tmp = a
    else if (m <= 1800000000000.0d0) then
        tmp = t_0
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -2.05e-107) {
		tmp = t_0;
	} else if (m <= -1.85e-264) {
		tmp = a;
	} else if (m <= 1800000000000.0) {
		tmp = t_0;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -2.05e-107:
		tmp = t_0
	elif m <= -1.85e-264:
		tmp = a
	elif m <= 1800000000000.0:
		tmp = t_0
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -2.05e-107)
		tmp = t_0;
	elseif (m <= -1.85e-264)
		tmp = a;
	elseif (m <= 1800000000000.0)
		tmp = t_0;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -2.05e-107)
		tmp = t_0;
	elseif (m <= -1.85e-264)
		tmp = a;
	elseif (m <= 1800000000000.0)
		tmp = t_0;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.05e-107], t$95$0, If[LessEqual[m, -1.85e-264], a, If[LessEqual[m, 1800000000000.0], t$95$0, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.05e-107 or -1.84999999999999998e-264 < m < 1.8e12

   1. Initial program 96.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 59.3

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 55.7

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 55.7%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -2.05e-107 < m < -1.84999999999999998e-264

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 96.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   7. Step-by-step derivation

   8. Simplified 70.9%

      \[\leadsto \color{blue}{a} \]

   if 1.8e12 < m

   1. Initial program 69.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 2.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right) + a} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot k\right) \cdot -10} + a \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(k \cdot -10\right)} + a \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

      5. *-lowering-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

   8. Simplified 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

       2. *-lowering-*.f64 23.2

          \[\leadsto -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

   11. Simplified 23.2%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-107}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq -1.85 \cdot 10^{-264}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1800000000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.23:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.23)
   (/ a (* k k))
   (if (<= m 5.8e-5)
     (/ a (+ (* k (+ k 10.0)) 1.0))
     (fma k (* a (fma 100.0 k -10.0)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.23) {
		tmp = a / (k * k);
	} else if (m <= 5.8e-5) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.23)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 5.8e-5)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.23], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.8e-5], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.23000000000000001

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 36.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 57.9

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 57.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -0.23000000000000001 < m < 5.8e-5

   1. Initial program 93.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 91.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

      4. +-lowering-+.f64 91.5

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

   7. Applied egg-rr 91.5%

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right) + 1}} \]

   if 5.8e-5 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 3.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 3.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1}{10}}, 1\right)} \]
   9. Step-by-step derivation

   10. Simplified 3.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right) + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) - 10 \cdot a, a\right)} \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{100 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a, a\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(100 \cdot k\right) \cdot a} + -10 \cdot a, a\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       9. accelerator-lowering-fma.f64 33.2

          \[\leadsto \mathsf{fma}\left(k, a \cdot \color{blue}{\mathsf{fma}\left(100, k, -10\right)}, a\right) \]

   13. Simplified 33.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 60.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.58:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.58)
   (/ a (* k k))
   (if (<= m 5.8e-5)
     (/ a (fma k (+ k 10.0) 1.0))
     (fma k (* a (fma 100.0 k -10.0)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.58) {
		tmp = a / (k * k);
	} else if (m <= 5.8e-5) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.58)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 5.8e-5)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.58], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.8e-5], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.57999999999999996

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 36.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 57.9

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 57.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -0.57999999999999996 < m < 5.8e-5

   1. Initial program 93.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 91.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   if 5.8e-5 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 3.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 3.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1}{10}}, 1\right)} \]
   9. Step-by-step derivation

   10. Simplified 3.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right) + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) - 10 \cdot a, a\right)} \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{100 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a, a\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(100 \cdot k\right) \cdot a} + -10 \cdot a, a\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       9. accelerator-lowering-fma.f64 33.2

          \[\leadsto \mathsf{fma}\left(k, a \cdot \color{blue}{\mathsf{fma}\left(100, k, -10\right)}, a\right) \]

   13. Simplified 33.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.58:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.25:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.25)
   (/ a (* k k))
   (if (<= m 4e-7) (/ a (fma k k 1.0)) (fma k (* a (fma 100.0 k -10.0)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.25) {
		tmp = a / (k * k);
	} else if (m <= 4e-7) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = fma(k, (a * fma(100.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.25)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 4e-7)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = fma(k, Float64(a * fma(100.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.25], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4e-7], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(100.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.25

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 36.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 57.9

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 57.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -0.25 < m < 3.9999999999999998e-7

   1. Initial program 93.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 92.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k}, 1\right)} \]
   7. Step-by-step derivation

   8. Simplified 88.2%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k}, 1\right)} \]

   if 3.9999999999999998e-7 < m

   1. Initial program 70.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 3.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\frac{k \cdot k - 10 \cdot 10}{k - 10}}, 1\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k - 10 \cdot 10\right) \cdot \frac{1}{k - 10}}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(k \cdot k + \left(\mathsf{neg}\left(10 \cdot 10\right)\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, \mathsf{neg}\left(10 \cdot 10\right)\right)} \cdot \frac{1}{k - 10}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \mathsf{neg}\left(\color{blue}{100}\right)\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, \color{blue}{-100}\right) \cdot \frac{1}{k - 10}, 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1}{k - 10}}, 1\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{\color{blue}{k + \left(\mathsf{neg}\left(10\right)\right)}}, 1\right)} \]

      12. metadata-eval 3.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + \color{blue}{-10}}, 1\right)} \]

   7. Applied egg-rr 3.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, k, -100\right) \cdot \frac{1}{k + -10}}, 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1}{10}}, 1\right)} \]
   9. Step-by-step derivation

   10. Simplified 3.4%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{-0.1}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right) + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) - 10 \cdot a, a\right)} \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{100 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, 100 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a, a\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(100 \cdot k\right) \cdot a} + -10 \cdot a, a\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(100 \cdot k + -10\right)}, a\right) \]

       9. accelerator-lowering-fma.f64 33.3

          \[\leadsto \mathsf{fma}\left(k, a \cdot \color{blue}{\mathsf{fma}\left(100, k, -10\right)}, a\right) \]

   13. Simplified 33.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(100, k, -10\right), a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.55:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(99, k, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.55)
   (/ a (* k k))
   (if (<= m 4e-7) (/ a (fma k k 1.0)) (fma k (* a (fma 99.0 k -10.0)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.55) {
		tmp = a / (k * k);
	} else if (m <= 4e-7) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = fma(k, (a * fma(99.0, k, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.55)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 4e-7)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = fma(k, Float64(a * fma(99.0, k, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.55], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4e-7], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.55000000000000004

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 36.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 57.9

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 57.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -0.55000000000000004 < m < 3.9999999999999998e-7

   1. Initial program 93.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 92.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k}, 1\right)} \]
   7. Step-by-step derivation

   8. Simplified 88.2%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k}, 1\right)} \]

   if 3.9999999999999998e-7 < m

   1. Initial program 70.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 3.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

      4. +-lowering-+.f64 3.6

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)} + 1} \]

   7. Applied egg-rr 3.6%

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right) + 1}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a, a\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + -10 \cdot a, a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \left(\mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right) + -10 \cdot a, a\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(\left(a + -100 \cdot a\right)\right)\right) \cdot k} + -10 \cdot a, a\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + \left(\mathsf{neg}\left(-100 \cdot a\right)\right)\right)} \cdot k + -10 \cdot a, a\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \left(\color{blue}{-1 \cdot a} + \left(\mathsf{neg}\left(-100 \cdot a\right)\right)\right) \cdot k + -10 \cdot a, a\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(-1 \cdot a - -100 \cdot a\right)} \cdot k + -10 \cdot a, a\right) \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(a \cdot \left(-1 - -100\right)\right)} \cdot k + -10 \cdot a, a\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(\left(-1 - -100\right) \cdot k\right)} + -10 \cdot a, a\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, a \cdot \left(\left(-1 - -100\right) \cdot k\right) + \color{blue}{a \cdot -10}, a\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(\left(-1 - -100\right) \cdot k + -10\right)}, a\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot \left(\left(-1 - -100\right) \cdot k + -10\right)}, a\right) \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, a \cdot \color{blue}{\mathsf{fma}\left(-1 - -100, k, -10\right)}, a\right) \]

      17. metadata-eval 33.3

          \[\leadsto \mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(\color{blue}{99}, k, -10\right), a\right) \]

   10. Simplified 33.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(99, k, -10\right), a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.33:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1750000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.33)
   (/ a (* k k))
   (if (<= m 1750000000000.0) (/ a (fma k k 1.0)) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.33) {
		tmp = a / (k * k);
	} else if (m <= 1750000000000.0) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.33)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1750000000000.0)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.33], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1750000000000.0], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.330000000000000016

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 36.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 57.9

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 57.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -0.330000000000000016 < m < 1.75e12

   1. Initial program 93.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 88.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k}, 1\right)} \]
   7. Step-by-step derivation

   8. Simplified 85.2%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k}, 1\right)} \]

   if 1.75e12 < m

   1. Initial program 69.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 2.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right) + a} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot k\right) \cdot -10} + a \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(k \cdot -10\right)} + a \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

      5. *-lowering-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

   8. Simplified 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

       2. *-lowering-*.f64 23.2

          \[\leadsto -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

   11. Simplified 23.2%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.33:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1750000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1700000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.5e-14)
   (/ a (* k k))
   (if (<= m 1700000000000.0) (/ a (fma k 10.0 1.0)) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.5e-14) {
		tmp = a / (k * k);
	} else if (m <= 1700000000000.0) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.5e-14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1700000000000.0)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.5e-14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1700000000000.0], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.5000000000000001e-14

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 38.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. *-lowering-*.f64 59.2

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 59.2%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -6.5000000000000001e-14 < m < 1.7e12

   1. Initial program 93.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 89.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{1 + 10 \cdot k}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{10 \cdot k + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot 10} + 1} \]

      3. accelerator-lowering-fma.f64 62.0

         \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}} \]

   8. Simplified 62.0%

      \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}} \]

   if 1.7e12 < m

   1. Initial program 69.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 2.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right) + a} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot k\right) \cdot -10} + a \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(k \cdot -10\right)} + a \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

      5. *-lowering-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

   8. Simplified 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

       2. *-lowering-*.f64 23.2

          \[\leadsto -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

   11. Simplified 23.2%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1700000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1700000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1700000000000.0) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1700000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1700000000000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1700000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1700000000000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1700000000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1700000000000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1700000000000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.7e12

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 65.0

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   7. Step-by-step derivation

   8. Simplified 26.1%

      \[\leadsto \color{blue}{a} \]

   if 1.7e12 < m

   1. Initial program 69.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. +-lowering-+.f64 2.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right) + a} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot k\right) \cdot -10} + a \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(k \cdot -10\right)} + a \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

      5. *-lowering-*.f64 9.6

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

   8. Simplified 9.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

       2. *-lowering-*.f64 23.2

          \[\leadsto -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

   11. Simplified 23.2%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1700000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 20.2% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 87.0%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

   7. associate-*l* N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

   11. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. +-lowering-+.f64 43.9

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Simplified 43.9%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a} \]
7. Step-by-step derivation

8. Simplified 18.6%

   \[\leadsto \color{blue}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))