Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 99.7%

Time: 10.5s

Alternatives: 15

Speedup: 1.1×

• 21.9% 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: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 5 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{t\_0} + \frac{10}{t\_0}, k, \frac{1}{t\_0}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 5e-90)
     t_0
     (pow (fma (+ (/ k t_0) (/ 10.0 t_0)) k (/ 1.0 t_0)) -1.0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 5e-90) {
		tmp = t_0;
	} else {
		tmp = pow(fma(((k / t_0) + (10.0 / t_0)), k, (1.0 / t_0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 5e-90)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(k / t_0) + Float64(10.0 / t_0)), k, Float64(1.0 / t_0)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 5e-90], t$95$0, N[Power[N[(N[(N[(k / t$95$0), $MachinePrecision] + N[(10.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * k + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000019e-90

   1. Initial program 97.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.00000000000000019e-90 < k

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 79.4

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 79.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-90}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{{k}^{m} \cdot a} + \frac{10}{{k}^{m} \cdot a}, k, \frac{1}{{k}^{m} \cdot a}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{\left(-m\right)}}{a}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-81}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(10 + k\right) \cdot t\_0, k, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (pow k (- m)) a)))
   (if (<= k 2e-81) (* (pow k m) a) (/ 1.0 (fma (* (+ 10.0 k) t_0) k t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, -m) / a;
	double tmp;
	if (k <= 2e-81) {
		tmp = pow(k, m) * a;
	} else {
		tmp = 1.0 / fma(((10.0 + k) * t_0), k, t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ Float64(-m)) / a)
	tmp = 0.0
	if (k <= 2e-81)
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(1.0 / fma(Float64(Float64(10.0 + k) * t_0), k, t_0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[k, 2e-81], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(1.0 / N[(N[(N[(10.0 + k), $MachinePrecision] * t$95$0), $MachinePrecision] * k + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e-81

   1. Initial program 97.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.9999999999999999e-81 < k

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 79.2

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 79.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-81}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(10 + k\right) \cdot \frac{{k}^{\left(-m\right)}}{a}, k, \frac{{k}^{\left(-m\right)}}{a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 97.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1 < k

   1. Initial program 74.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 74.5

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 74.5

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in k around inf

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

   12. Applied rewrites 99.9%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -9e+14)
     t_0
     (if (<= m 5.6e-7) (/ 1.0 (fma (+ 10.0 k) (/ k a) (/ 1.0 a))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -9e+14) {
		tmp = t_0;
	} else if (m <= 5.6e-7) {
		tmp = 1.0 / fma((10.0 + k), (k / a), (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = t_0;
	elseif (m <= 5.6e-7)
		tmp = Float64(1.0 / fma(Float64(10.0 + k), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -9e+14], t$95$0, If[LessEqual[m, 5.6e-7], N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -9e14 or 5.60000000000000038e-7 < m

   1. Initial program 90.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -9e14 < m < 5.60000000000000038e-7

   1. Initial program 88.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 88.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 99.1%

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

Alternative 5: 66.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(10 + k\right) \cdot k\\ t_1 := t\_0 \cdot k\\ \mathbf{if}\;m \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;1 \cdot \frac{a}{\mathsf{fma}\left(t\_1 \cdot \left(10 + k\right), t\_0, 1\right)}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 10 + k, \left(-k\right) \cdot k\right) \cdot \frac{a}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (+ 10.0 k) k)) (t_1 (* t_0 k)))
   (if (<= m -1.8e+17)
     (* 1.0 (/ a (fma (* t_1 (+ 10.0 k)) t_0 1.0)))
     (if (<= m 1.35)
       (/ 1.0 (fma (+ 10.0 k) (/ k a) (/ 1.0 a)))
       (* (fma t_1 (+ 10.0 k) (* (- k) k)) (/ a 1.0))))))

C

double code(double a, double k, double m) {
	double t_0 = (10.0 + k) * k;
	double t_1 = t_0 * k;
	double tmp;
	if (m <= -1.8e+17) {
		tmp = 1.0 * (a / fma((t_1 * (10.0 + k)), t_0, 1.0));
	} else if (m <= 1.35) {
		tmp = 1.0 / fma((10.0 + k), (k / a), (1.0 / a));
	} else {
		tmp = fma(t_1, (10.0 + k), (-k * k)) * (a / 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(10.0 + k) * k)
	t_1 = Float64(t_0 * k)
	tmp = 0.0
	if (m <= -1.8e+17)
		tmp = Float64(1.0 * Float64(a / fma(Float64(t_1 * Float64(10.0 + k)), t_0, 1.0)));
	elseif (m <= 1.35)
		tmp = Float64(1.0 / fma(Float64(10.0 + k), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = Float64(fma(t_1, Float64(10.0 + k), Float64(Float64(-k) * k)) * Float64(a / 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(10.0 + k), $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * k), $MachinePrecision]}, If[LessEqual[m, -1.8e+17], N[(1.0 * N[(a / N[(N[(t$95$1 * N[(10.0 + k), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(10.0 + k), $MachinePrecision] + N[((-k) * k), $MachinePrecision]), $MachinePrecision] * N[(a / 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8e17

   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. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 66.7%

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

   if -1.8e17 < m < 1.3500000000000001

   1. Initial program 89.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 89.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 89.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 89.0

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

   4. Applied rewrites 89.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 98.7%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 96.6%

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

   if 1.3500000000000001 < m

   1. Initial program 83.5%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 34.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 53.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;1 \cdot \frac{a}{\mathsf{fma}\left(\left(\left(\left(10 + k\right) \cdot k\right) \cdot k\right) \cdot \left(10 + k\right), \left(10 + k\right) \cdot k, 1\right)}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(10 + k\right) \cdot k\right) \cdot k, 10 + k, \left(-k\right) \cdot k\right) \cdot \frac{a}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a - \frac{\frac{a}{k} \cdot -99}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(10 + k\right) \cdot k\right) \cdot k, 10 + k, \left(-k\right) \cdot k\right) \cdot \frac{a}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (/ (- a (/ (* (/ a k) -99.0) k)) (* k k))
   (if (<= m 1.35)
     (/ 1.0 (fma (+ 10.0 k) (/ k a) (/ 1.0 a)))
     (* (fma (* (* (+ 10.0 k) k) k) (+ 10.0 k) (* (- k) k)) (/ a 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = (a - (((a / k) * -99.0) / k)) / (k * k);
	} else if (m <= 1.35) {
		tmp = 1.0 / fma((10.0 + k), (k / a), (1.0 / a));
	} else {
		tmp = fma((((10.0 + k) * k) * k), (10.0 + k), (-k * k)) * (a / 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(Float64(a - Float64(Float64(Float64(a / k) * -99.0) / k)) / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64(1.0 / fma(Float64(10.0 + k), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(10.0 + k) * k) * k), Float64(10.0 + k), Float64(Float64(-k) * k)) * Float64(a / 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(N[(a - N[(N[(N[(a / k), $MachinePrecision] * -99.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(10.0 + k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(10.0 + k), $MachinePrecision] + N[((-k) * k), $MachinePrecision]), $MachinePrecision] * N[(a / 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   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. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{a - \frac{-99 \cdot \frac{a}{k}}{k}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.5%

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

   if -9e14 < m < 1.3500000000000001

   1. Initial program 89.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.9

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 88.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.6%

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

   if 1.3500000000000001 < m

   1. Initial program 83.5%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 34.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 53.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a - \frac{\frac{a}{k} \cdot -99}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(10 + k\right) \cdot k\right) \cdot k, 10 + k, \left(-k\right) \cdot k\right) \cdot \frac{a}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(10 + k\right) \cdot k\right) \cdot k, 10 + k, \left(-k\right) \cdot k\right) \cdot \frac{a}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (* (/ 1.0 (* k k)) a)
   (if (<= m 1.35)
     (/ 1.0 (fma (+ 10.0 k) (/ k a) (/ 1.0 a)))
     (* (fma (* (* (+ 10.0 k) k) k) (+ 10.0 k) (* (- k) k)) (/ a 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = (1.0 / (k * k)) * a;
	} else if (m <= 1.35) {
		tmp = 1.0 / fma((10.0 + k), (k / a), (1.0 / a));
	} else {
		tmp = fma((((10.0 + k) * k) * k), (10.0 + k), (-k * k)) * (a / 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(Float64(1.0 / Float64(k * k)) * a);
	elseif (m <= 1.35)
		tmp = Float64(1.0 / fma(Float64(10.0 + k), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(10.0 + k) * k) * k), Float64(10.0 + k), Float64(Float64(-k) * k)) * Float64(a / 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 1.35], N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(10.0 + k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(10.0 + k), $MachinePrecision] + N[((-k) * k), $MachinePrecision]), $MachinePrecision] * N[(a / 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{1}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if -9e14 < m < 1.3500000000000001

   1. Initial program 89.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.9

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 88.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.6%

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

   if 1.3500000000000001 < m

   1. Initial program 83.5%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 34.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 53.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(10 + k\right) \cdot k\right) \cdot k, 10 + k, \left(-k\right) \cdot k\right) \cdot \frac{a}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (* (/ 1.0 (* k k)) a)
   (if (<= m 8e-5)
     (/ 1.0 (fma (+ 10.0 k) (/ k a) (/ 1.0 a)))
     (fma (fma (* 99.0 a) k (* -10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = (1.0 / (k * k)) * a;
	} else if (m <= 8e-5) {
		tmp = 1.0 / fma((10.0 + k), (k / a), (1.0 / a));
	} else {
		tmp = fma(fma((99.0 * a), k, (-10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(Float64(1.0 / Float64(k * k)) * a);
	elseif (m <= 8e-5)
		tmp = Float64(1.0 / fma(Float64(10.0 + k), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = fma(fma(Float64(99.0 * a), k, Float64(-10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 8e-5], N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * a), $MachinePrecision] * k + N[(-10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{1}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if -9e14 < m < 8.00000000000000065e-5

   1. Initial program 88.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 88.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 99.1%

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

   if 8.00000000000000065e-5 < m

   1. Initial program 83.8%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.5

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

   5. Applied rewrites 3.5%

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-980 \cdot a, k, 99 \cdot a\right), k, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

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

Alternative 9: 60.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (* (/ 1.0 (* k k)) a)
   (if (<= m 8e-5)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (fma (* 99.0 a) k (* -10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = (1.0 / (k * k)) * a;
	} else if (m <= 8e-5) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma((99.0 * a), k, (-10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(Float64(1.0 / Float64(k * k)) * a);
	elseif (m <= 8e-5)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(fma(Float64(99.0 * a), k, Float64(-10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 8e-5], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * a), $MachinePrecision] * k + N[(-10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{1}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if -9e14 < m < 8.00000000000000065e-5

   1. Initial program 88.7%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 8.00000000000000065e-5 < m

   1. Initial program 83.8%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.5

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

   5. Applied rewrites 3.5%

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-980 \cdot a, k, 99 \cdot a\right), k, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 85000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (* (/ 1.0 (* k k)) a)
   (if (<= m 85000000000.0) (/ a (fma (+ 10.0 k) k 1.0)) (* (* -10.0 k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = (1.0 / (k * k)) * a;
	} else if (m <= 85000000000.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(Float64(1.0 / Float64(k * k)) * a);
	elseif (m <= 85000000000.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 85000000000.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{1}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if -9e14 < m < 8.5e10

   1. Initial program 88.2%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 8.5e10 < m

   1. Initial program 84.2%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 4.9%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 13.7%

       \[\leadsto \left(-10 \cdot k\right) \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 85000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 85000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (/ a (* k k))
   (if (<= m 85000000000.0) (/ a (fma (+ 10.0 k) k 1.0)) (* (* -10.0 k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = a / (k * k);
	} else if (m <= 85000000000.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 85000000000.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 85000000000.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   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. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 57.5%

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

   if -9e14 < m < 8.5e10

   1. Initial program 88.2%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 8.5e10 < m

   1. Initial program 84.2%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 4.9%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 13.7%

       \[\leadsto \left(-10 \cdot k\right) \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 85000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 86000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2e-60)
   (/ a (* k k))
   (if (<= m 86000000000.0) (/ a (fma 10.0 k 1.0)) (* (* -10.0 k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-60) {
		tmp = a / (k * k);
	} else if (m <= 86000000000.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2e-60)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 86000000000.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2e-60], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 86000000000.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.9999999999999999e-60

   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. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 60.3%

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

   if -1.9999999999999999e-60 < m < 8.6e10

   1. Initial program 87.3%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 65.1%

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

   if 8.6e10 < m

   1. Initial program 84.2%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 4.9%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 13.7%

       \[\leadsto \left(-10 \cdot k\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1.05 \cdot 10^{-285}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -1.05e-285) t_0 (if (<= k 0.1) (fma (* -10.0 a) k a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -1.05e-285) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -1.05e-285)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.05e-285], t$95$0, If[LessEqual[k, 0.1], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -1.04999999999999992e-285 or 0.10000000000000001 < k

   1. Initial program 83.3%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 38.3%

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

   if -1.04999999999999992e-285 < k < 0.10000000000000001

   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. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 53.4%

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

   12. Applied rewrites 53.4%

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

Alternative 14: 25.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 86000000000:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 86000000000.0) (* 1.0 a) (* (* -10.0 k) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 86000000000.0) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 86000000000.0d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((-10.0d0) * k) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 86000000000.0) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 86000000000.0:
		tmp = 1.0 * a
	else:
		tmp = (-10.0 * k) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 86000000000.0)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(-10.0 * k) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 86000000000.0)
		tmp = 1.0 * a;
	else
		tmp = (-10.0 * k) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 86000000000.0], N[(1.0 * a), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.6e10

   1. Initial program 92.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in m around 0

      \[\leadsto 1 \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 34.0%

      \[\leadsto 1 \cdot a \]

   if 8.6e10 < m

   1. Initial program 84.2%

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

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 4.9%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 13.7%

       \[\leadsto \left(-10 \cdot k\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 19.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* 1.0 a))

C

double code(double a, double k, double m) {
	return 1.0 * 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 = 1.0d0 * a
end function

Java

public static double code(double a, double k, double m) {
	return 1.0 * a;
}

Python

def code(a, k, m):
	return 1.0 * a

Julia

function code(a, k, m)
	return Float64(1.0 * a)
end

MATLAB

function tmp = code(a, k, m)
	tmp = 1.0 * a;
end

Wolfram

code[a_, k_, m_] := N[(1.0 * a), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 84.2

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

5. Applied rewrites 84.2%

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

6. Taylor expanded in m around 0

   \[\leadsto 1 \cdot a \]
7. Step-by-step derivation

8. Applied rewrites 22.8%

   \[\leadsto 1 \cdot a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))