Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 98.2%

Time: 10.4s

Alternatives: 14

Speedup: 1.1×

• 22.2% 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) INFINITY)
     (/ t_0 (fma (+ k 10.0) k 1.0))
     (* (* k (* k k)) (* a (fma k 1e-6 -1e-5))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= ((double) INFINITY)) {
		tmp = t_0 / fma((k + 10.0), k, 1.0);
	} else {
		tmp = (k * (k * k)) * (a * fma(k, 1e-6, -1e-5));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= Inf)
		tmp = Float64(t_0 / fma(Float64(k + 10.0), k, 1.0));
	else
		tmp = Float64(Float64(k * Float64(k * k)) * Float64(a * fma(k, 1e-6, -1e-5)));
	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[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   4. Applied egg-rr 98.0%

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

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

   10. Simplified 1.6%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{{k}^{2}}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot {k}^{2}\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)} \]

       8. *-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right) \]

       10. *-commutative N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       13. lower-fma.f64 100.0

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 43.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, 99, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_0 2e-298)
     (/ a (* k (+ k 10.0)))
     (if (<= t_0 1e+287)
       (/ a (fma k 10.0 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma k (* a (fma k 99.0 -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_0 <= 2e-298) {
		tmp = a / (k * (k + 10.0));
	} else if (t_0 <= 1e+287) {
		tmp = a / fma(k, 10.0, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma(k, (a * fma(k, 99.0, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e-298)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (t_0 <= 1e+287)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(k, Float64(a * fma(k, 99.0, -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[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-298], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+287], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(k * 99.0 + -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))) < 1.99999999999999982e-298

   1. Initial program 97.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 45.1

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

   5. Simplified 45.1%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 35.9

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

   8. Simplified 35.9%

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

   if 1.99999999999999982e-298 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.0000000000000001e287

   1. Initial program 99.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 76.5%

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

   if 1.0000000000000001e287 < (/.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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.7

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

   5. Simplified 3.7%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 49.9

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

   8. Simplified 49.9%

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

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around 0

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

      1. +-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. lower-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. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, a \cdot \left(k \cdot \left(-1 \cdot \color{blue}{-99}\right) + -10\right), a\right) \]

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{-99}\right), -10\right), a\right) \]

      18. metadata-eval 79.2

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

   10. Simplified 79.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, 99, -10\right), a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, 99, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, 99, -10\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 1e+287)
       (/ a (fma k 10.0 1.0))
       (if (<= t_1 INFINITY) t_0 (fma k (* a (fma k 99.0 -10.0)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+287) {
		tmp = a / fma(k, 10.0, 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(k, (a * fma(k, 99.0, -10.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+287)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(k, Float64(a * fma(k, 99.0, -10.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+287], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(k * N[(a * N[(k * 99.0 + -10.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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 or 1.0000000000000001e287 < (/.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 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 39.4

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

   5. Simplified 39.4%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 39.5

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

   8. Simplified 39.5%

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

   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))) < 1.0000000000000001e287

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 73.1%

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

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around 0

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

      1. +-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. lower-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. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, a \cdot \left(k \cdot \left(-1 \cdot \color{blue}{-99}\right) + -10\right), a\right) \]

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{-99}\right), -10\right), a\right) \]

      18. metadata-eval 79.2

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

   10. Simplified 79.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, a \cdot \mathsf{fma}\left(k, 99, -10\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 1e+287)
       (/ a (fma k 10.0 1.0))
       (if (<= t_1 INFINITY) t_0 (fma a (* k -10.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+287) {
		tmp = a / fma(k, 10.0, 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(a, (k * -10.0), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+287)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(a, Float64(k * -10.0), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+287], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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 or 1.0000000000000001e287 < (/.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 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 39.4

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

   5. Simplified 39.4%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 39.5

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

   8. Simplified 39.5%

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

   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))) < 1.0000000000000001e287

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 73.1%

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

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 1.6

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

   5. Simplified 1.6%

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

   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-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 34.3

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

   8. Simplified 34.3%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ t_2 := \mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-298}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))))
        (t_2 (fma a (* k -10.0) a)))
   (if (<= t_1 2e-298)
     t_0
     (if (<= t_1 1e+287) t_2 (if (<= t_1 INFINITY) t_0 t_2)))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	double t_2 = fma(a, (k * -10.0), a);
	double tmp;
	if (t_1 <= 2e-298) {
		tmp = t_0;
	} else if (t_1 <= 1e+287) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	t_2 = fma(a, Float64(k * -10.0), a)
	tmp = 0.0
	if (t_1 <= 2e-298)
		tmp = t_0;
	elseif (t_1 <= 1e+287)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-298], t$95$0, If[LessEqual[t$95$1, 1e+287], t$95$2, If[LessEqual[t$95$1, Infinity], t$95$0, t$95$2]]]]]]

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))) < 1.99999999999999982e-298 or 1.0000000000000001e287 < (/.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 97.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 39.6

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

   5. Simplified 39.6%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 39.8

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

   8. Simplified 39.8%

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

   if 1.99999999999999982e-298 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.0000000000000001e287 or +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 52.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 53.1

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

   5. Simplified 53.1%

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

   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-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 55.5

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

   8. Simplified 55.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{-298}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) INFINITY)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (* (* k (* k k)) (* a (fma k 1e-6 -1e-5)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= ((double) INFINITY)) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = (k * (k * k)) * (a * fma(k, 1e-6, -1e-5));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= Inf)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(Float64(k * Float64(k * k)) * Float64(a * fma(k, 1e-6, -1e-5)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 98.0

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.0

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

   4. Applied egg-rr 98.0%

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

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 1.6

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

   5. Simplified 1.6%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 1.6

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in k around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

   10. Simplified 1.6%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{{k}^{2}}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot {k}^{2}\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)} \]

       8. *-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right) \]

       10. *-commutative N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       13. lower-fma.f64 100.0

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 22.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) 0.0)
   (/ a (* k 10.0))
   (fma a (* k -10.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 0.0) {
		tmp = a / (k * 10.0);
	} else {
		tmp = fma(a, (k * -10.0), a);
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 44.8

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

   5. Simplified 44.8%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 35.5

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

   8. Simplified 35.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 15.9

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

   11. Simplified 15.9%

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

   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. Initial program 73.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 32.7

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

   5. Simplified 32.7%

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

   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-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 35.6

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

   8. Simplified 35.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.1%

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

Alternative 8: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -4.6e-5)
     t_0
     (if (<= m 1.02e-44) (/ a (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 (m <= -4.6e-5) {
		tmp = t_0;
	} else if (m <= 1.02e-44) {
		tmp = a / 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 (m <= -4.6e-5)
		tmp = t_0;
	elseif (m <= 1.02e-44)
		tmp = Float64(a / 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[m, -4.6e-5], t$95$0, If[LessEqual[m, 1.02e-44], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -4.6e-5 or 1.0199999999999999e-44 < m

   1. Initial program 89.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Simplified 100.0%

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

   if -4.6e-5 < m < 1.0199999999999999e-44

   1. Initial program 94.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 94.0

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

   5. Simplified 94.0%

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

Alternative 9: 82.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k \cdot k\right)\\ \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{100}{k \cdot t\_0}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (* k k))))
   (if (<= m -3.7e+20)
     (* a (/ 100.0 (* k t_0)))
     (if (<= m 1.65)
       (/ a (fma k (+ k 10.0) 1.0))
       (* t_0 (* a (fma k 1e-6 -1e-5)))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k * k);
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (100.0 / (k * t_0));
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0 * (a * fma(k, 1e-6, -1e-5));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k * k))
	tmp = 0.0
	if (m <= -3.7e+20)
		tmp = Float64(a * Float64(100.0 / Float64(k * t_0)));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(t_0 * Float64(a * fma(k, 1e-6, -1e-5)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -3.7e+20], N[(a * N[(100.0 / N[(k * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.7e20

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 32.7

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

   5. Simplified 32.7%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 42.7

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

   8. Simplified 42.7%

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

   9. Taylor expanded in k around inf

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

       1. metadata-eval N/A

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

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

          \[\leadsto \frac{\color{blue}{\left(a - -100 \cdot \frac{a}{{k}^{2}}\right)} - 10 \cdot \frac{a}{k}}{{k}^{2}} \]

       3. associate--r+ N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 64.8%

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

   12. Taylor expanded in k around 0

       \[\leadsto \color{blue}{100 \cdot \frac{a}{{k}^{4}}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{100 \cdot a}{{k}^{4}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot 100}}{{k}^{4}} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{a \cdot \frac{100}{{k}^{4}}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \frac{100}{{k}^{4}}} \]

       5. lower-/.f64 N/A

          \[\leadsto a \cdot \color{blue}{\frac{100}{{k}^{4}}} \]

       6. metadata-eval N/A

          \[\leadsto a \cdot \frac{100}{{k}^{\color{blue}{\left(3 + 1\right)}}} \]

       7. pow-plus N/A

          \[\leadsto a \cdot \frac{100}{\color{blue}{{k}^{3} \cdot k}} \]

       8. lower-*.f64 N/A

          \[\leadsto a \cdot \frac{100}{\color{blue}{{k}^{3} \cdot k}} \]

       9. cube-mult N/A

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

       10. unpow2 N/A

           \[\leadsto a \cdot \frac{100}{\left(k \cdot \color{blue}{{k}^{2}}\right) \cdot k} \]

       11. lower-*.f64 N/A

           \[\leadsto a \cdot \frac{100}{\color{blue}{\left(k \cdot {k}^{2}\right)} \cdot k} \]

       12. unpow2 N/A

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

       13. lower-*.f64 72.8

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

   14. Simplified 72.8%

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

   if -3.7e20 < m < 1.6499999999999999

   1. Initial program 94.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 93.3

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

   5. Simplified 93.3%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

   10. Simplified 48.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{{k}^{2}}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot {k}^{2}\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)} \]

       8. *-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right) \]

       10. *-commutative N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       13. lower-fma.f64 80.2

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right) \]

   13. Simplified 80.2%

       \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{100}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e+24)
   (/ a (* k k))
   (if (<= m 1.65)
     (/ a (fma k (+ k 10.0) 1.0))
     (* (* k (* k k)) (* a (fma k 1e-6 -1e-5))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e+24) {
		tmp = a / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (k * (k * k)) * (a * fma(k, 1e-6, -1e-5));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e+24)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(k * Float64(k * k)) * Float64(a * fma(k, 1e-6, -1e-5)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e+24], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000001e24

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 33.1

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

   5. Simplified 33.1%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.7

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

   8. Simplified 60.7%

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

   if -6.8000000000000001e24 < m < 1.6499999999999999

   1. Initial program 94.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 92.2

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

   5. Simplified 92.2%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

   10. Simplified 48.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{{k}^{2}}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot {k}^{2}\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)} \]

       8. *-commutative N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right) \]

       10. *-commutative N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)} \]

       13. lower-fma.f64 80.2

           \[\leadsto \left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right) \]

   13. Simplified 80.2%

       \[\leadsto \color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot -1 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e+24)
   (/ a (* k k))
   (if (<= m 1.65)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (* (* k (* k k)) -1e-5)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e+24) {
		tmp = a / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * ((k * (k * k)) * -1e-5);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e+24)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(k * k)) * -1e-5));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e+24], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * -1e-5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000001e24

   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 33.1

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

   5. Simplified 33.1%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.7

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

   8. Simplified 60.7%

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

   if -6.8000000000000001e24 < m < 1.6499999999999999

   1. Initial program 94.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 92.2

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

   5. Simplified 92.2%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

   10. Simplified 48.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{-1}{100000} \cdot \left(a \cdot {k}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{100000} \cdot a\right) \cdot {k}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \frac{-1}{100000}\right)} \cdot {k}^{3} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{100000} \cdot {k}^{3}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{100000} \cdot {k}^{3}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{100000} \cdot {k}^{3}\right)} \]

       6. cube-mult N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right) \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \]

       10. lower-*.f64 66.0

           \[\leadsto a \cdot \left(-1 \cdot 10^{-5} \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \]

   13. Simplified 66.0%

       \[\leadsto \color{blue}{a \cdot \left(-1 \cdot 10^{-5} \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot -1 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.15 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot -1 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.15e-30)
   (/ a (* k k))
   (if (<= m 1.65) (/ a (fma k 10.0 1.0)) (* a (* (* k (* k k)) -1e-5)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.15e-30) {
		tmp = a / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = a * ((k * (k * k)) * -1e-5);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.15e-30)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(k * k)) * -1e-5));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.15e-30], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * -1e-5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.14999999999999983e-30

   1. Initial program 98.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 35.9

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

   5. Simplified 35.9%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.1

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

   8. Simplified 59.1%

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

   if -2.14999999999999983e-30 < m < 1.6499999999999999

   1. Initial program 95.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 95.1

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

   5. Simplified 95.1%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 60.4%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-/.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. lower-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. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

   10. Simplified 48.1%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{-1}{100000} \cdot \left(a \cdot {k}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{100000} \cdot a\right) \cdot {k}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \frac{-1}{100000}\right)} \cdot {k}^{3} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{100000} \cdot {k}^{3}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{100000} \cdot {k}^{3}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{100000} \cdot {k}^{3}\right)} \]

       6. cube-mult N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right) \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto a \cdot \left(\frac{-1}{100000} \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \]

       10. lower-*.f64 66.0

           \[\leadsto a \cdot \left(-1 \cdot 10^{-5} \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \]

   13. Simplified 66.0%

       \[\leadsto \color{blue}{a \cdot \left(-1 \cdot 10^{-5} \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.15 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot -1 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 20.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, k \cdot -10, a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (fma a (* k -10.0) a))

C

double code(double a, double k, double m) {
	return fma(a, (k * -10.0), a);
}

Julia

function code(a, k, m)
	return fma(a, Float64(k * -10.0), a)
end

Wolfram

code[a_, k_, m_] := N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. lower-/.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. lower-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. distribute-rgt-in N/A

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

   15. *-lft-identity N/A

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

   16. associate-*l* N/A

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

   17. lft-mult-inverse N/A

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

   18. metadata-eval N/A

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

   19. lower-+.f64 41.6

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

5. Simplified 41.6%

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

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-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.6

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

8. Simplified 19.6%

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

Alternative 14: 20.0% 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 91.1%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. lower-/.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. lower-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. distribute-rgt-in N/A

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

   15. *-lft-identity N/A

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

   16. associate-*l* N/A

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

   17. lft-mult-inverse N/A

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

   18. metadata-eval N/A

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

   19. lower-+.f64 41.6

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

5. Simplified 41.6%

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

6. Taylor expanded in k around 0

   \[\leadsto \frac{a}{\color{blue}{1}} \]
7. Step-by-step derivation

8. Simplified 17.1%

   \[\leadsto \frac{a}{\color{blue}{1}} \]
9. Step-by-step derivation

   1. /-rgt-identity 17.1

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

10. Applied egg-rr 17.1%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))